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Found 121 results

1. ## Nested Snippets (snippets within snippets)

With the new features in Alfred Snippets, I'm considering consolidating all my snippets from aText into Alfred. One feature that I use often in aText is the ability to embed a snippet inside another snippet expansion. This allows me to treat smaller snippets as variables that can be reused in other larger snippets. Is this possible with Alfred's Snippets?
2. ## Feature Request : Interactive Variables for Snippets

Not sure if this has been requested, but I was dissapointed to see that the "Snippets" feature was not touched in this last 4.0 update. I would love to be able to create snippets that interacted and requested info based on how you define it. For instance, if my snippet is defined as: "Your name is %name%", when executed, a prompt will request the %name% value and then paste the shortcut --> "Your name is Christian". I am trying to use RocketTypist to do this but it's buggy, this would be a nice addition to the snippet feature in Alfred, which I have found to be very responsive. Thanks!
3. ## Feature Request: Rich Text in Snippets and Clipboard History

I know people have requested image support in the clipboard history feature in Alfred. That would be great, but I would really like to see support for maintaining text formatting (rich text) in the snippets and clipboard history. As of now all text is converted into plain text in the history and snippets. It would great if formatting were not lost.
4. ## Newer emojis in the Emoji Snippets Pack

Hi there, I really wanted to be able to use :ok_hand: as a shortcut for 👌, but the emoji pack didn't support these new emojis... So I made my own! There ae more than 400 emojis that ware added, it still works the same And I'll keep updating it for the forseeable future. Download link: https://github.com/paris-ci/Alfred-Emoji-Pack-Update/raw/master/snippets/Emoji Pack Update.alfredsnippets Code used to generate the pack: https://github.com/paris-ci/Alfred-Emoji-Pack-Update It's basically the same thing as before, so here is the installation tutorial : http://joelcalifa.com/blog/alfred-emoji-snippet-pack/ (Don't use the download link there, it's still the old version, I am not the original author)
5. ## iOS keyboard text expansion

This forum has seen a lot of posts recently (including my own) which are basically geared towards allowing Alfred to replace TextExpander in the wake of the Smile pricing/sync kerfuffle -- ideas like allowing tabs/form filling, input within snippet expansion, snippets triggering workflows/scripts, case sensitivity in snippets, etc. Thinking about what it would take for me to be able to replace the 250+ snippets I have in TE with Alfred snippets, there are two killer features....1) is snippets to trigger scripts (mostly for date math)....I suspect this is in the works....2) is the ability to use my snippets on iOS. As far as I've seen, this hasn't come up, but it seems like an area where Alfred could really raise the bar. Smile has a pretty widely adopted API ( https://smilesoftware.com/textexpander-ios/apps) but the TE keyboard itself is pretty slow and generic. So I'd like to see an Alfred iOS keyboard, either as part of Alfred Remote or a separate app. It could just do snippet expansion on iOS using Alfred's synced snippets. If it could run scripts, this would be even better. And for the icing on the cake, maybe it could communicate with Alfred on OS X, run a query and return the results (the kind of two-way communication I think people were hoping to see emerge with Alfred Remote). OR....now that Smile has lowered their price on TE 6 to $20/year for existing users, and$40/year or so for new users, maybe it would make sense to not invest development resources into snippets. For me at least (as an existing user) the revised TE pricing makes switching a lot less urgent and compelling -- the main concern is the need to rely on Smile's sync service. Hence the subject line....if Alfred is going to go all in with snippets and replace TextExpander, it seems like an iOS keyboard is going to be necessary for many users....if that's not on the roadmap, it might not be worth prioritizing snippet features. Thoughts?
6. ## Import TextExpander snippets into Alfred v3

Is there any way to import over my TextExpander snippets into Alfred v3? I found this, but Github is a bit beyond me, no idea how to get started.
7. ## 2 quick Alfred snippet questions

Using Alfred 4 + Powerpack... I want to migrate from another text expander and keep everything in Alfred. Is there a way to require a delimiter such as a space, period, question mark, etc. in order to autoexpand a snippet? Other text expanders do that. Is there a way to create a shortcut to open the Snippets window directly without having to access Alfred Preferences and then select "Snippets" from the sidebar? I would like a fast way to make a selection into a snippet. Thanks!
8. ## Input Fields for Snippet Expansion

Hi alfredders I love Alfred and struggle to find anything wrong with it. It has managed to substitute lots of tiny utility apps (weather, calculator, clipboard manager... Just to name the first that come to mind). For my line of work I type frequently the same text over and over and over. I've been using Typinator in the past and love its Input Field options (see attached screenshots). I was wondering if there was something similar in development for future version of Alfred or - EVEN BETTER - if a workflow already exists to achieve a similar result. In short I'd like to have the ability to expand a snippet with multiple choice to pick from (either single option or multiple choice, doesn't matter really). Or should i simply accept the idea that this is not achievable for the time being/future? I'm not a developer: I'm in the other side of the spectrum (read Designer) so not sure what I'm talking about here Thanks in advance everyone and forgive me if this was not the right section to post this!
9. ## Comprehensive Snippet Expansion for Latex Greek Letters

If you are a STEM student, you might have been familiar with the markup language LaTex. Typing equations in LaTex is painful and it is almost impossible to compile the latex document without getting an error once or twice. To facilitate writing LaTex documents, I have created a comprehensive (more than 2000) snippets especially dedicated to latex greek letters. Example usage: Type this: \\a This expands to: \alpha Type this: \\ax This expands to: $\alpha$ and so on. Here: prefix = \\ letter = a expands to = alpha The snippet can be found in my GitHub: https://github.com/bhishanpdl/ALFRED/blob/master/AlfredSnippets/LatexGreek.alfredsnippets Installation: 1. Download the snippet and double click 2. UNCHECK the option Strip snippet of auto expand flag (by default it is checked, uncheck it) Detailed usage: Some rules: 1. I call Partial, Angle and Bold as prefix terms. 2. I call other attributes such as bar, prime, hat, etc suffix. 3. x means surround by the two dollar sign. E.g to create $\alpha$ we need to type \\ax. 4. suffix attributes need only one extra letter at the beginning. Eg. to create \bar{\alpha} we need to type \\ab` where b means bar. 5. prefix attributes needs two surrounding letters. E.g. to create \partial{\alpha} we need to type \\pay here prefix p means partial and we also need last letter y.
10. ## Lost Snippets

Hi, I accidentally deleted my main collection of snippets, instead of just the one individual snippet inside that collection (wasn't paying attention to which '+/-' I was pressing). I don't have a sync location set and unfortunately don't have a time machine backup - does anyone know where the alfdb file goes when it's deleted? Not in trash, hoping it's still recoverable. Thanks in advance, any help is really appreciated!
11. ## [SOLVED] How to write the output of snippets to a text file and quickview it?

There are some ways to create and use the snippets in Alfred. 1. Using snip command . We can only see the command name, and copy the snippets but cant see the contents. 2. Using clipboard manager we can search and see the contents of snippets and copy it, but can not see the preview (eg. using shift key) Here, I would like to get help creating a workflow which will write the snippets to a file and quickview it. idea: command: qsnip hello effect: copies the snippet called hello and paste the contents "hello world" to a file called "mysnip.txt" in the directory of workflow. effect: qlmanage -p mysnip.txt Problem: I am unware how to save the snippet to the file "mysnip.txt" I have shared the minimal workflow here: https://github.com/bhishanpdl/Shared/blob/master/Alfred_questions/qsnip.alfredworkflow?raw=true
12. ## Undo deletion of Snippet collection

Hi all, I accidentally deleted the whole collection instead of one snippet. Is there a way to retrieve that? Maybe something on the file system in the Library folder? Thanks in advance for your help. - Charlie
13. ## Import snippets from other apps?

Would love to have a way to import snippets already created in other apps into Alfred? I'm thinking apps like aText, TextExpander, etc?
14. ## solved Snippet option will pop up while typing in a normal text field

Hi there, Recently, I noticed that when I'm typing some snippet keyword will be picked up by computer, even I'm not within the Alfred box. Eg, I have snippet as 'name', when I was typing inside Bear notes, when I type name, the snippet content will pop up into the text field. Can you please look into this? Ian
15. ## How to create a workflow that creates a file, then uses a snippet to populate it?

I'm trying to work out how to complete a second step in a new workflow that will streamline a workflow I use several times a day. I create Markdown notes throughout my day. I have a pretty basic workflow that runs a terminal command that, in turn, creates a file and opens it in my text editor. I then invoke a snippet that populates the note with predefined text, and multiple clipboard items that I save before I invoke the workflow. In other words: Save multiple bits of information to my Alfred clipboard in a predetermined sequence Run a terminal command through a simple workflow that creates the Markdown file, and opens it in my text editor When the file opens in my editor, I manually run a snippet that populates the file, including the clipboard items using {clipboard:0}, {clipboard:1}, etc syntax. The terminal command looks something like this: touch ~/path/to/directory/$(date +%Y-%m-%d)_Feedback_{query}_Interactions.md & open ~/path/to/directory/$(date +%Y-%m-%d)_Feedback_{query}_Interactions.md -a Atom The {query} is something I add when I trigger the workflow. The snippet contains predefined text with placeholders for my clipboard items. I'd like to combine all of that from into a single workflow. What I can't work out is how to include the 3rd step in the workflow so I can run it all as a single step. I don't suppose anyone has any suggestions?
16. ## Snippets Reformat Paragraphs in Microsoft Word

I have tried multiple different settings to fix it, but whenever I expand a snippet into Word, it reformats the paragraph spacing, messing with my document. Is there an option (in either Word or Alfred) to fix this? Thanks!
17. ## Minor GUI tweak for Snippets category

This is very minor, but also maybe very easy to adjust? In the Snippets section, can the ‘add’ and ‘remove’ buttons be oriented with the left side of their respective fields? It’s silly, but I click 'add collection’ all the time meaning to click ‘add snippet’ because it’s visually near to the field, and since we read left to right, I think orienting the GUI that way improves the flow. Plus, if they were both oriented to the left it would be less distance to travel between mouse clicks; microseconds in efficiency!
18. ## Text expansion issue when working in VS Code

I noticed that when I use text expansion in VS Code, the snippets don't expand fully. It seems to occur when the Markdown All in One extension is enabled. I've opened an issue here: Text snippets don't expand correctly when using text expansion, with Markdown All in One enabled · Issue #200 · neilsustc/vscode-markdown. I did some additional testing with TextExpander and I wasn't able to reproduce the issue when I expanded snippets in TextExpander, when the Markdown All in One extension is enabled in VS Code. Would there be something about how Alfred expands snippets that differs from how TE handles text expansion, that could be causing this issue in VS Code? I've only really noticed this issue in VS Code, and when the Markdown All in One extension is enabled. I've been using Atom as my primary editor and don't have this issue there, so this seems to be caused by some sort of interaction between VS Code (with the extension enabled), and Alfred. I don't suppose you have any insights that might be helpful in troubleshooting this?
19. ## Rich Text in Snippets

Is there a way to use rich text in snippets such as links or formatted text?
20. ## Cannot Enable Alfred Snippets

Hi When I click on 'Automatically expand snippets by keyword' the Alfred Text Service dialog is coming up but when I click on 'Let's get started!' then nothing happens. System Preferences is not coming nor is the dialog box to ask for permission. If I go to the Security & Privacy / Privacy tab there is nothing shown. I, eventually, was able to fix it by using 'Path Finder', go into Alfred3 package content and drag the Alfred Text Service into the Security & Privacy / Privacy tab.
21. ## Hotkey to quickly edit existing snippet

Hi guys, I use snippets a lot in Alfred 3, and I badly need a quicker way to edit existing snippets. They're so awesome! Please see 1st screenshot: if Cmd+S on selected snippet would open it directly (2nd screenshot) within Alfred preferences, that would be a simple but fantastic improvement! I'm suggesting Cmd+S for this because it is very similar to the already existing Cmd+S shortcut which transforms a clipping from clipboard history into a new snippet. Basically hitting Cmd+S would mean: "create a new snippet or open it for editing if it already exists". That would feel very intuitive! After hitting Cmd+S, this window would popup to make you able to quickly update the existing snippet. Thanks! Vermeer
22. ## Any luck with integration with Anki?

Hi all, I'm trying to use snippets to insert symbols (up arrow, down arrow, etc.) into anki cards on a Mac. My goal is to be able to insert them when making Anki cards but I am clueless on where to begin. Can someone point me in the right direction on where to learn? Thanks!
23. ## Allow for HTML content (hyperlinks) in Snippets

I was hoping to use Alfred's snippets to store and edit 'canned responses' across various email accounts I manage. It works beautifully for text, but any hyperlinks are dropped. That is, "click here for help" becomes "click here for help" I know in Gmail that copy/paste retains hyperlink data, but any copy/paste into a text editor will drop the hyperlinks (I assume this has to do with meta-something). I'm not even sure if the TextExpander tools feature something like this, but support for rich html snippets would be amazing!
24. ## Countdown delta

Hi. Trying to create an Alfred snippet that calculates a delta value from 'now' to a specific time or date. In short, I'd like to put a hard date in a snippet, and every time I insert it, the result is the number of 'days' or 'hours' between now and that hard date or time. Thoughts? Thanks!
25. ## snippets Allow to assign Snippets group to different application

It'll be great useful if we can use snippet groups only when some specific applications activated.
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Decomposition of finite algebras over finite fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:26:19Z http://mathoverflow.net/feeds/question/61533 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61533/decomposition-of-finite-algebras-over-finite-fields Decomposition of finite algebras over finite fields Franz Lemmermeyer 2011-04-13T10:51:56Z 2011-04-20T17:19:53Z
Let \$K\$ be a number field, \$Z_K\$ its ring of integers, and \$p\$ a rational prime number. Then \$A_p = Z_K/(p)\$ is a finite \${\mathbb F}_p\$-algebra. Using ideal arithmetic in \$Z_K\$ and the Chinese remainder theorem it is easily checked that \$A_p\$ is the direct sum of finite fields if \$p\$ is unramified, and has additonal nil-rings as components if \$p\$ is ramified.

This decomposition result looks simple enough to have a direct and not too complicated proof. Basing it on the close relation between \$A_p\$ and \${\mathbb F}_p[X]/(f)\$, where \$f\$ is the minimal polynomial of a generator of \$K\$, brings in problems with primes dividing the discriminant of \$f\$. Thus let me state my main question explicitly:

Is there a simple proof that \$A_p\$ is a direct sum of finite fields and some easily described nil rings?

In addition, I'd be grateful for pointers to the relevant literature, in particular to classification theorems of which the result above is a special case.
 http://mathoverflow.net/questions/61533/decomposition-of-finite-algebras-over-finite-fields/61534#61534 Answer by Charles Matthews for Decomposition of finite algebras over finite fields Charles Matthews 2011-04-13T11:04:52Z 2011-04-13T11:04:52Z
The general form of the answer is easy to anticipate: a finite commutative ring is artinian. It will be a product of finite local rings, of characteristic that is a prime number. So your question is really which local rings you get in this case. I believe a certain amount can be done by linear algebra.
 http://mathoverflow.net/questions/61533/decomposition-of-finite-algebras-over-finite-fields/62437#62437 Answer by David Tweedle for Decomposition of finite algebras over finite fields David Tweedle 2011-04-20T17:19:53Z 2011-04-20T17:19:53Z
A good starting point, at least from a number theory perspective, is the book "Local fields" by J.P. Serre. Chapter 1, Section 5 deals with the question you ask. It says that if \$\$ pZ_K = \prod_{\mathfrak{p}|p} \mathfrak{p}^{e_\mathfrak{p}}, \$\$ then \$\$ A_p \cong \prod_{\mathfrak{p}|p} Z_K/\mathfrak{p}^{e_\mathfrak{p}}. \$\$ The nil-rings are exactly described by the prime ideals \$\mathfrak{p}\$ of \$Z_K\$ which have \$e_\mathfrak{p}>1\$. I think that this book is very useful, but I'm sure that this topic is also dealt with in algebraic number theory textbooks, or standard algebra textbooks.

A more general situation is:

If we assume that \$L,K\$ are fields with \$[L:K]=n\$, \$L/K\$ separable, \$A\subset K\$, with fraction field equal to \$K\$, and \$A\$ is a Dedekind domain. Further assume that the integral closure of \$A\$ in \$L\$, say \$B\subset L\$, is also a Dedekind domain. Then the proof of the fact you want relies on the factorization of \$pB\$ as a product of prime ideals in \$B\$, and the Chinese remainder theorem or "Approximation Lemma" (in Serre's book). For a better explanation, read Serre's book.

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ISSN : 1598-6721(Print)
ISSN : 2288-0771(Online)
The Korean Society of Manufacturing Process Engineers Vol.19 No.6 pp.23-28
DOI : https://doi.org/10.14775/ksmpe.2020.19.06.023

# Methods to Obtain Approximate Responses of a Non-Linear Vibration Isolation System

Gun-Myung Lee*#
*Research Center for Aircraft Parts Technology, Gyeongsang National University
Corresponding Author : gmlee@gnu.ac.kr Tel: +82-55-772-1621, Fax: +82-55-772-1577
14/02/2020 18/03/2020 20/03/2020

## Abstract

A non-linear vibration isolation system composed of a non-linear spring and a linear damper was presented in a previous study. The advantage of the proposed isolator is the simple structure of the system. When the base of the isolator is harmonically excited, the response component of the mass at the excitation frequency was approximated using three different methods: linear approximation, harmonic balance, and higher-order frequency response functions (FRFs). The method using higher-order FRFs produces significantly more accurate results compared with the other methods. The error between the exact and approximate responses does not increase monotonously with the excitation amplitude and is less than 2%.

# 비선형 진동절연 시스템의 근사적 응답을 구하는 방법

이 건명*#
*경상대학교 항공기부품기술연구소

## 초록

This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 1. Introduction

Vibration isolation is a procedure by which the undesirable effects of vibration are reduced. It involves the insertion of a resilient member (or isolator) between the vibrating mass and the source of vibration to achieve a reduction in the dynamic response of the system under specified vibration excitation[1]. The suspension system of a car seat is an example of a vibration isolator.

To resolve the conflict between small transmissibility and large static deflection, the use of non-linear springs is proposed. A vibration isolator with a non-linear spring was proposed and its characteristics investigated in previous research[2]. The proposed non-linear spring is composed of two symmetric linear springs depicted in Fig. 1. The structure of the spring is very simple compared to other non-linear springs(e.g., disc springs[3]), and the force-displacement relationship can be expressed easily, resulting in a straightforward dynamic analysis of systems equipped with the spring.

If the non-linear relationship between the spring force and the displacement can be represented by a polynomial, the equation of motion of the vibration isolator with this non-linear spring can be solved approximately.

In this study, the equation of motion of the non-linear isolator was solved approximately using three different methods: linear approximation, harmonic balance, and higher-order frequency response functions(FRFs), and the results were compared with numerical integration results which are assumed to be exact.

## 2. Description of Non-linear Equationsolving Methods

### 2.1 Harmonic balance method

Because the harmonic balance method[4] is based on an analysis primarily applicable to periodic functions, it is particularly useful for computing approximations to periodic solutions. In this method the steady-state response of a non-linear system subjected to a harmonic input is expressed by a sum of harmonics. In a simple case the response is expressed as

$y ( t ) = A cos ω t$
(1)

The expression for the response is inserted into a non-linear equation, and by equating coefficients of each harmonic on both sides the amplitude of each harmonic is obtained.

### 2.2 Higher-order FRFs

When a system with polynomial non-linearities is subjected to harmonic excitation

$x ( t ) = X cos ω t = X 2 e j ω t + X 2 e − j ω t ,$
(2)

the response can be expressed using higher-order FRFs as follows. Detailed explanation on the Volterra series can be found in References [5,6].

$y ( t ) = H 1 ( ω ) X 2 e j ω t + H 1 ( − ω ) X 2 e − j ω t + H 2 ( ω , ω ) ( X 2 ) 2 e j 2 ω t + 2 H 2 ( ω , − ω ) ( X 2 ) 2 + H 2 ( − ω , − ω ) ( X 2 ) 2 e − j 2 ω t + H 3 ( ω , ω , ω ) ( X 2 ) 3 e j 3 ω t + 3 H 3 ( ω , ω , − ω ) ( X 2 ) 3 e j ω t + 3 H 3 ( ω , − ω , − ω ) ( X 2 ) 3 e − j ω t + H 3 ( − ω , − ω , − ω ) ( X 2 ) 3 e − j 3 ω t + ⋯$
(3)

where $H n ( ω , ⋯ , ω , − ω , ⋯ , − ω )$ is a higher-order FRF. The arguments of the FRF are composed of ω and - ω. Denoting the FRF by Hn,i when ω is repeated n - i times and - ω is repeated i times in the argument list, the above response can be expressed as:

$y ( t ) = ∑ n = 1 ∞ y n ( t ) = ∑ n = 1 ∞ ∑ i = 0 n ( n i ) H n , i ( X 2 ) n e j ( n − 2 i ) ω t$
(4)

where $( n i )$ is a binomial coefficient equal to $n ! / i ! ( n − i ) !$. Denote the complex ratio of the response component at the excitation frequency ω to the excitation component at the same frequency as $H ^ 1 ( ω )$. By considering only the terms associated with the input $X 2 e j ω t$ in Eq. (3), an estimate of $H ^ 1 ( ω )$for a non-linear system can be obtained:

$H ^ 1 ( ω ) = Y ( ω ) X ( ω ) = H 1 ( ω ) + ( 3 1 ) H 3 ( ω , ω , − ω ) ( X 2 ) 2 + ( 5 2 ) H 5 ( ω , ω , ω , − ω , − ω ) ( X 2 ) 4 + ⋯ = ∑ m = 0 ∞ ( 2 m + 1 m ) H 2 m + 1 , m ( X 2 ) 2 m$
(5)

The equation of motion of a single degree-of-freedom system with cubic stiffness subjected to harmonic excitation can be written

$m y ¨ ( t ) + c y ˙ ( t ) + k 1 y ( t ) + k 2 y ( t ) 2 + k 3 y ( t ) 3 = X 2 ( e j ω t + e − j ω t )$
(6)

Substituting Eq. (3) into the above equation and equating the coefficients of the $( X 2 ) e j ω t$ terms on both sides, one can obtain

$H 1 ( ω ) = 1 k 1 − m ω 2 + j c ω$
(7)

Similarly, equating the coefficients of the $( X 2 ) 2 e j 2 ω t$ terms, one can obtain

$( − 4 m ω 2 + j c 2 ω + k 1 ) H 2 ( ω , ω ) + k 2 H 1 ( ω ) 2 = 0$
(8)

$H 2 ( ω , ω ) = − k 2 H 1 ( ω ) 2 k 1 − m ( 2 ω ) 2 + j c 2 ω = − k 2 H 1 ( ω ) 2 H 1 ( 2 ω )$
(9)

Since a higher-order FRF is a complex valued function, it can be represented by magnitude and phase. Fig. 2 shows $H 2 ( ω , ω )$ for a non-linear system which will be considered later.

Extending this method of harmonic probing[7], all higher-order FRFs in Eq. (5) can be obtained. Based on this result a higher-order FRF can be expressed in terms of lower-order FRFs. After all the FRFs in Eq. (5) are calculated, the response component at the excitation frequency, Y(ω) can be obtained from Eq. (5).

## 3. Application of the Solving Methods

### 3.1 Proposed non-linear isolator

The proposed non-linear isolator is composed of two symmetric linear springs, a linear damper, and a base as depicted in Fig. 3. The linear springs have spring constant k, free length l0, and are inclined by θ from the horizontal base. When the upper support of the springs moves downward by y, the vertical force applied on the support from the springs can be calculated easily. Fig. 4 illustrates the variation of the spring force with y for k = 10,000 N/m, l0 = 0.7 m, and θ = 0.7752 rad(44.42°). The spring force reaches a maximum value of 1261 N at y = 0.239 m. The figure shows that the stiffness of the spring decreases with the displacement of the support and becomes negative for large displacements.

When the base of the vibration isolator is excited as depicted in Fig. 3, the motion of the mass is described by the following equation.

$m y ¨ = − c ( y ˙ − x ˙ ) − f ( y − x )$
(10)

In Eq. (10) a dot on a letter represents differentiation with respect to time. c and f(x) represent the damping constant and the spring force, respectively. The excitation of the base x(t) is given by X sin ωt. The displacement of the mass, y(t) is measured from the equilibrium position due to gravity. Letting z = y - x, Eq. (10) becomes

$m z ¨ + c z ˙ + f ( z ) = m ω 2 X sin ω t$
(11)

Solving Eq. (11) for z numerically and adding x, the response of the mass is obtained. Computing a Fourier Transform of y(t), its frequency component at the excitation frequency ω is obtained. The considered system is equipped with the above non-linear spring and has the parameters : m = 100 kg, c = 1000 Ns/m, ω = 10 rad/s.

### 3.2 Approximate solutions

The response of the mass for the harmonic excitation of the base was calculated by solving the equation of motion numerically. By computing a Fourier transform of the response, the response component at the excitation frequency ω is obtained, which is assumed to be exact. The amplitude of the response component at the excitation frequency was also calculated approximately and the results compared based on three methods : linear approximation, harmonic balance, and higher-order FRFs.

#### 3.2.1 Linear approximation

At the static equilibrium of the system, the deflection of the mass and slope of the tangent to the curve in Fig. 4 are found to be 0.13 m and 5028 N/m, respectively. If the non-linear spring is approximated by a linear spring with this stiffness, the ratio of the amplitude of the mass to that of the base becomes 1.0022 based on linear theory. Thus the amplitude of the mass is Y = 0.0501 m for X = 0.05 m. Comparing this amplitude with the exact response component 0.049328 m, the error is 1.57%. Because the system is approximated as a linear system, the ratio does not change with the excitation amplitude. The response component of the mass at the excitation frequency and the error were calculated repeatedly for various excitation amplitudes. As the excitation amplitude increases, the error increases monotonously and reaches 40.65% for X = 0.22 m as depicted in Fig. 5.

#### 3.2.2 Harmonic balance method

In Eq. (11), f(z) represents the non-linear spring force. If the origin of the coordinate axes is set to the static equilibrium position and the spring force around the origin is represented by a cubic polynomial $f ( y ) = k 1 y + k 2 y 2 + k 3 y 3$, the coefficients are k1 = 5031.6 N/m, k2 = -21126 N/m2, and k3 = -13715 N/m3 based on a regression analysis in the range 0 ≤ y ≤ 0.239 m as depicted in Fig. 4. Then Eq. (11) can be written as

$m z ¨ + c z ˙ + k 1 z + k 2 z 2 + k 3 z 3 = m ω 2 X sin ( ω t + ϕ )$
(12)

In Eq. (12) the unknown phase ϕ is added to the excitation to obtain a single-term harmonic response z(t) = Z sin ωt[8].

Inserting z(t) into Eq. (12), the following equation is obtained.

$− m ω 2 Z sin ω t + c ω Z cos ω t + k 1 Z sin ω t + k 2 Z 2 sin 2 ω t + k 3 Z 3 sin 3 ω t = m ω 2 X sin ( ω t + ϕ )$
(13)

Using the following trigonometric relations

$sin 2 ω t = 1 2 − 1 2 cos 2 ω t$
(14)

$sin 3 ω t = 3 4 sin ω t − 1 4 sin3 ω t$
(15)

$sin ( ω t + ϕ ) = sin ω t cos ϕ + cos ω t sin ϕ$
(16)

and equating the coefficients of sinωt and cosωt terms on both sides, the following equations are obtained.

$( k 1 − m ω 2 ) Z + 3 4 k 3 Z 3 = m ω 2 X cos ϕ$
(17)

$c ω Z = m ω 2 X sin ϕ$
(18)

Solving these equations simultaneously produces Z = 0.04474 m and ϕ = 2.03356 rad. Because z(t) = Z sin ωt is the response for the excitation 2X sin (ωt + ϕ), the response for the excitation 2X sin ωt would be z(t) = Z sin (ωt - ϕ). Adding x(t) to z(t) , y(t) is obtained with an amplitude of the response component at the excitation frequency of 0.050044 m, resulting in an error of 1.45% which is slightly less than that for linear approximation. As the excitation amplitude increases, the error increases monotonously and reaches 36.41% for X = 0.22 m as depicted in Fig. 4. The error varies with the excitation amplitude similarly to, but slightly less than the case of linear approximation. If more harmonics are included for the solution of Eq. (12), more accurate responses could be obtained. However, the procedure is very complicated as depicted in Reference [8]. Therefore, an approximate solution with more harmonics has not been attempted in this research.

#### 3.2.3 Higher-order FRFs

If the spring force in Eq. (11) is approximated by a cubic polynomial, the equation of motion is expressed as

$m z ¨ + c z ˙ + k 1 z + k 2 z 2 + k 3 z 3 = m ω 2 X sin ω t$
(19)

The response component at the excitation frequency was obtained using higher-order FRFs, as explained in Section 2.2, and x(t) was added to obtain the frequency component of y(t) . Increasing the maximum order of higher-order FRFs in Eq. (5), the above calculation was repeated. For the maximum order n ≥ 7, the obtained results converged. Consequently the maximum order n = 9 was used for future calculations. The response component at the excitation frequency was 0.049367 m for X = 0.05 m, with an error of 0.079%. Comparing the result with the previous two cases, this method produces more accurate results. As the excitation amplitude increases, the error does not increase rapidly as in the previous cases. The error becomes a maximum of 1.53% when X = 0.2 m as depicted in Fig. 5. This error is significantly smaller than those obtained by the two previous methods. When the excitation amplitude is increased beyond X = 0.22 m, the response diverges.

## 4. Conclusion

A non-linear vibration isolator composed of a non-linear spring and a linear damper was proposed in previous research.

When the base of the isolator is excited harmonically, the response component of the mass at the excitation frequency was calculated approximately using three different methods: linear approximation, harmonic balance, and higher-order FRFs. For the linear approximation method the error between the exact and approximate responses increases monotonously with the excitation amplitude. For the harmonic balance method the error varies with the excitation amplitude similarly to, but slightly less than the case of linear approximation. The method using higher-order FRFs produces significantly more accurate results compared to the other methods. The error does not increase monotonously with the excitation amplitude, and is less than 2%.

## Figure

Proposed non-linear spring
Magnitude and phase of H2(ω,ω) of a non-linear system
Considered vibration isolation system
Variation of the spring force with the deflection
Variation of the error with the excitation amplitude for linear approximation(dash-dot line), harmonic balance(dashed line), and higher-order FRFs(solid line)

## Reference

1. Rao, S. S., Mechanical Vibrations(6th edition), Pearson Education, pp. 851-883, 2018.
2. Lee, G.-M. and Park, H. S., “Characteristics of a Vibration Isolation System with Non-linear Springs,” Transactions of the Korean Society for Noise and Vibration Engineering, Vol. 27, No. 7, pp. 844~849, 2017.
3. Shin, D. H., Lee, J. Y. and Oh, J.-E., “Prediction of Load-displacement of the Disc Spring with the Friction,” Transactions of the Korean Society for Noise and Vibration Engineering, Vol. 22, No. 4, pp. 344~351, 2012.
4. Lee, G.-M., “Approximate Response of a Non-linear Vibration Isolation System Using the Harmonic Balance Method,” Journal of the Korean Society of Manufacturing Process Engineers, Vol. 17, No. 6, pp. 124~129, 2018.
5. Tomlinson, G. R., Manson, G. and Lee, G.-M., “A Simple Criterion for Establishing an Upper Limit to the Harmonic Excitation Level of the Duffing Oscillator Using the Volterra Series,” Journal of Sound and Vibration, Vol. 190, No. 5, pp. 751~762, 1996.
6. Lee, G.-M., “Estimation of Non-linear System Parameters Using Higher-order Frequency Response Functions,” Mechanical Systems and Signal Processing, Vol. 11, No. 2, pp. 219~228, 1997.
7. Bedrosian, E. and Rice, S. O., “The Output Properties of Volterra Systems Driven by Harmonic and Gaussian Inputs,” Proceedings of the IEEE, Vol. 59, No. 12, pp. 1688~1707, 1971.
8. Hamdan, M. N. and Burton, T. D., “On the Steady State Response and Stability of Non-linear Oscillators Using Harmonic Balance,” Journal of Sound and Vibration, Vol. 166, No. 2, pp. 255~266, 1993.


text:
## Fields

• A field is a set $𝔽$ with functions  $\begin{array}{ccc}𝔽×𝔽& \to & 𝔽\\ \left({c}_{1},{c}_{2}\right)& ⟼& {c}_{1}+{c}_{2}\end{array}$     and     $\begin{array}{ccc}𝔽×𝔽& \to & 𝔽\\ \left({c}_{1},{c}_{2}\right)& ⟼& {c}_{1}{c}_{2}\end{array}$,
(a)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then $\left({c}_{1}+{c}_{2}\right)+{c}_{3}=\left({c}_{1}+\left({c}_{2}+{c}_{3}\right)$,
(b)   If ${c}_{1},{c}_{2}\in 𝔽$ then ${c}_{1}+{c}_{2}={c}_{2}+{c}_{1}$,
(c)   There exists a zero, $0\in 𝔽$, such that if $c\in 𝔽$ then $0+c=c+0=c$,
(d)   If $c\in 𝔽$ then there exists an additive inverse to $c$, $-c\in 𝔽$, such that $\left(-c\right)+c=c+\left(-c\right)=0$,
(e)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then ${c}_{1}\left({c}_{2}{c}_{3}\right)=\left({c}_{1}{c}_{2}\right){c}_{3}$,
(f)   If ${c}_{1},{c}_{2}\in 𝔽$ then ${c}_{1}{c}_{2}={c}_{2}{c}_{1}$,
(g)   There exists an identity, $1\in 𝔽$, such that $1\ne 0$ and if $c\in 𝔽$ then $1c=c1=c$,
(h)   If $c\in 𝔽$ and $c\ne 0$ then there exists an inverse (sometimes called a multiplicative inverse), ${c}^{-1}\in 𝔽$, such that $c{c}^{-1}={c}^{-1}c=1$,
(i)   If ${c}_{1},{c}_{2},{c}_{3}\in 𝔽$ then ${c}_{1}\left({c}_{2}+{c}_{3}\right)={c}_{1}{c}_{2}+{c}_{1}{c}_{3}$.
• A subfield of a field $𝔽$ is a subset $𝕂\subseteq 𝔽$ such that
(a)   If ${k}_{1},{k}_{2}\in 𝕂$ then ${k}_{1}+{k}_{2}\in 𝕂$,
(b)   $0\in 𝕂$,
(c)   If $k\in 𝕂$ then $-k\in 𝕂$,
(d)   If ${k}_{1},{k}_{2}\in 𝕂$ then ${k}_{1}{k}_{2}\in 𝕂$,
(e)   $1\in 𝕂$,
(f)   If $k\in 𝕂$ then ${k}^{-1}\in 𝕂$,

Note that every field is a commutative ring and the only conditions in the defnition of a field that are not in the definition of a ring are (f) and (h).

Important examples of fields are:

(a)   The rational numbers $ℚ$, the real numbers $ℝ$, and the complex numbers $ℂ$,
(b)   $ℤ/pℤ$, where $p$ is a prime.

#### Homomorphisms

Field homomorphisms might be used to compare fields. The only problem is that there aren't many interesting field homomorphisms, as we show in Proposition (fldhominj). We shall study fields in more depth in Part V.

Let $𝕂$ and $𝔽$ be fields with identities ${1}_{𝕂}$ and ${1}_{𝔽}$, respectively.

• A field homomorphism from $𝕂$ to $𝔽$ is a function $f:𝕂\to 𝔽$ such that
(a)   If ${k}_{1},{k}_{2}\in 𝕂$ then $f\left({k}_{1}+{k}_{2}\right)=f\left({k}_{1}\right)+f\left({k}_{2}\right)$,
(b)   If ${k}_{1},{k}_{2}\in 𝕂$ then $f\left({k}_{1}{k}_{2}\right)=f\left({k}_{1}\right)f\left({k}_{2}\right)$,
(c)   $f\left({1}_{𝕂}\right)=f\left({1}_{𝔽}\right)$.

HW: Show that if $f:𝕂\to 𝔽$ is a field homomorphism then $f\left({0}_{𝕂}\right)={0}_{𝔽}$, where ${0}_{𝕂}$ and ${0}_{𝔽}$ are the zeros in $𝕂$ and $𝔽$, respectively.
HW: Explain why conditions (a) and (b) in the definition of a field homomorphism do not imply condition (c).

If $f:𝕂\to 𝔽$ is a field homomorphism then $f$ is injective.

Proposition (fldhominj) stated another way, says that the kernel of any field homomorphism is $\left\{0\right\}$. This means that we cannot get an interesting analogue of Theorem ??? for fields. Proposition (fldhominj) also shows that if $f:𝕂\to 𝔽$ is a field homomorphism then $\mathrm{im}f=f\left(𝕂\right)$ is a subfield of $𝔽$.

## Notes and References

These notes are written to highlight the analogy between groups and group actions, rings and modules, and fields and vector spaces.

## References

[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.


text:
## Iterative solution of strongly nonsymmetric systems of linear algebraic equations.(English. Russian original)Zbl 0946.65018

Comput. Math. Math. Phys. 37, No. 11, 1241-1251 (1997); translation from Zh. Vychisl. Mat. Mat. Fiz. 37, No. 11, 1283-1293 (1997).
It is well known that most of the widely accepted preconditioners are designed to improve the properties of the symmetric part of system $$Au=b$$. It often happens that difficulties are associated with the ill-conditioned skew-symmetric part of the system to be solved. The preconditioner suggested in this paper allows one to solve the problem in the simplest and most natural way. The authors examine the convergence of Richardson’s method as applied to the preconditioned system in the case when the original matrix is dissipative. Sufficient conditions for the convergence of the method are presented, a procedure for finding the optimal value of the iteration parameter is developed, and an estimate for the convergence rate is given.

### MSC:

 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling


text:
# Physics 434, 2016: Homework 9

Back to the main Teaching page.

Please turn on the assignment either as a PDF file to me by email, or as a printout/writeup to my mailbox in physics. Detailed derivations (with explanations) and calculations must be present for the problems for full credit. Submit your code if a problem requires you to program.

3. How much information can a spiking neuron transmit? This is limited from above by its entropy rate. Let's represent a neuron as releasing action potentials with a Poisson process with a certain rate ${\displaystyle r}$, and let's calculate the entropy rate of the Poisson process. First represent this process by discretizing time in intervals ${\displaystyle \Delta t}$. Explain why the entropy of the Poisson generated sequence of duration ${\displaystyle T}$ (or, alternatively, ${\displaystyle n=T/\Delta t}$ symbols) is exactly proportional to time, that is ${\displaystyle S=sn}$, where ${\displaystyle s}$ is some constant. Thus we only need to calculate the entropy of a single symbol, this ${\displaystyle s}$, in order to find the entropy rate as ${\displaystyle R={\frac {sT}{\Delta tT}}}$. Does this rate have a finite value as ${\displaystyle \Delta t\to 0}$? Why or why not? Estimate the maximum bitrate of a neuron that can control placement of its spikes to the accuracy of 1 ms.
3. How much information can a spiking neuron transmit? This is limited from above by its entropy rate. Let's represent a neuron as releasing action potentials with a Poisson process with a certain rate ${\displaystyle r}$, and let's calculate the entropy rate of the Poisson process. First represent this process by discretizing time in intervals ${\displaystyle \Delta t}$. Explain why the entropy of the Poisson generated sequence of duration ${\displaystyle T}$ (or, alternatively, ${\displaystyle n=T/\Delta t}$ symbols) is exactly proportional to time, that is ${\displaystyle S=sn}$, where ${\displaystyle s}$ is some constant. Thus we only need to calculate the entropy of a single symbol, this ${\displaystyle s}$, in order to find the entropy rate as ${\displaystyle R={\frac {sT}{\Delta tT}}}$. Does this rate have a finite value as ${\displaystyle \Delta t\to 0}$? Why or why not? Estimate the maximum bitrate of a neuron that can control placement of its spikes to the accuracy of 1 ms.
4. Suppose now the neuron has what's called a refractory period. That is, after a spike, a neuron cannot fire for the time ${\displaystyle \tau _{r}}$. What is the entropy rate of such neuron?


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Question

# $$50\ g\ CaCO_{3}$$ are heated to $$1073\ K$$ in a $$5L$$ vessel. What percentage of $$CaCO_{3}$$ would decompose at equilibrium? The value of $$K_{p}$$ for the reaction.$$CaCO_{3}(s) \rightleftharpoons CaO(s) + CO_{2}(g)$$ is $$1.15\ atm$$ at $$1073\ K$$.

Solution

## Amount of $$CaCO_3 = 50 g$$Molar mass of $$CaCO_3 = 40 + 12 \times 3 \times 16 = 100 gm$$number of mole of $$CaCO_3 = \dfrac{50}{100} = 0.5 mole$$$$CaCO_3 \rightleftharpoons CaO + CO_2 (g)$$$$k_p = P_{CO_2} = 1.15 atm$$volume = $$52$$ temp =  $$1073 k$$$$n = \dfrac{P_{CO_2} v}{RT} = \dfrac{1.15 \times 5 }{0.0821 \times 1073} = 0.065$$$$\%$$ of $$CaCO_3$$ decompose = $$\dfrac{0.065}{5} \times 100$$$$= 1.3\%$$Chemistry

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text:
# [NTG-context] xtable

Meer, H. van der H.vanderMeer at uva.nl
Fri Jun 22 12:00:40 CEST 2012

On 22 jun. 2012, at 10:53, Wolfgang Schuster wrote:

>
> Am 21.06.2012 um 16:27 schrieb Meer, H. van der:
>
>> Can one put individual horizontal and vertical lines between rows/columns in xtable? Just like for example \HL in \starttable?
>
> You can use MetaPost to draw the rules but it’s awkward because you can make a setup like this
>
>  \setuxptable[column][1][background=…]
>
> which is only supported by natural tables. It would be simpler when stables had a counter like \currentxtablecolumn
> which could be accessed to make a test like “if \currentxtablecolumn = 1 then … fi” but this isn’t the case.

I see. No hope here.

But I found some sort of solution which I communicate here. It is bit of a kludge but does more or less what I had in mind. The following macros do the work in my xml=>xtable module:
<table frameon="t,b"> also <thead> <tr> <td> and so on.

\def\setframeparts#1#2#3{%
\doifinset{t}{#1}{#3[topframe=#2]}%
\doifinset{b}{#1}{#3[bottomframe=#2]}%
\doifinset{l}{#1}{#3[leftframe=#2]}%
\doifinset{r}{#1}{#3[rightframe=#2]}%
}
..#2 being \setupxtable or \setupframed, just as needed
\doifnot{\xmlatt{#1}{frameon}}{\empty}%
{#2[frame=off]\setframeparts{\xmlatt{#1}{frameon}}{on}{#2}}%
\doifnot{\xmlatt{#1}{frameoff}}{\empty}%
{#2[frame=on]\setframeparts{\xmlatt{#1}{frameoff}}{off}{#2}}%
..

>
>> The workings of parameter spaceinbetween on \startxtable remain a mystery to me. Can someone enlighten me?
>
> It inserts a space between each row, the amount of space is the same as \blank[…] would insert.
>

Do I conclude correctly that the dimension of the space is fixed to what \blank[] does?
I experimented with various values, but did not get much out of it.

Hans van der Meer


text:
+0

i dot know math at all how do i solve a fraction

0
46
1

hellllllllllllppuuuhh meeeee

May 18, 2022

#1
+1768
+1

Dividing is the same as multiplying a value by its reciprocal. The reciprocal is the value when you swap the numerator and the denominator. For example, the reciprocal of $$7 \over 9$$ is $$9 \over 7$$.

This means that $${5 \over 8} \div {7 \over 9} = {5 \over 8 } \times {9 \over 7}$$. Notice how the first fraction is the same, and ONLY the second fraction was converted into a reciprocal.

Can you take it from here?

May 18, 2022


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# Rapid synchronized fabrication of vascularized thermosets and composites

## Abstract

Bioinspired vascular networks transport heat and mass in hydrogels, microfluidic devices, self-healing and self-cooling structures, filters, and flow batteries. Lengthy, multistep fabrication processes involving solvents, external heat, and vacuum hinder large-scale application of vascular networks in structural materials. Here, we report the rapid (seconds to minutes), scalable, and synchronized fabrication of vascular thermosets and fiber-reinforced composites under ambient conditions. The exothermic frontal polymerization (FP) of a liquid or gelled resin facilitates coordinated depolymerization of an embedded sacrificial template to create host structures with high-fidelity interconnected microchannels. The chemical energy released during matrix polymerization eliminates the need for a sustained external heat source and greatly reduces external energy consumption for processing. Programming the rate of depolymerization of the sacrificial thermoplastic to match the kinetics of FP has the potential to significantly expedite the fabrication of vascular structures with extended lifetimes, microreactors, and imaging phantoms for understanding capillary flow in biological systems.

## Introduction

Biological materials possess hierarchical vascular networks that mediate heat and mass transport in response to external and internal stimuli, enabling complex living systems to thrive in extreme environments1,2. The replication of such pervasive vascular networks in engineered tissues and organoids for improving cellular proliferation3,4,5, studying fluid flow6, and pneumatic actuation7 has garnered significant interest in therapeutics, prosthetics, and soft robotics3,4,5,6. Vascular and porous synthetic materials are also used for electrical insulation in microelectronics8, gas exchange in synthetic devices9,10, thermal management in heat exchangers and fiber-reinforced composites11,12, chemical reactions in flow batteries and microreactors13,14,15, and material regeneration in self-healing structures16,17.

The new vasculature in biological load-bearing materials such as bone is created through coordinated deposition of new tissue and removal of old tissue, triggered by coupled cellular processes18,19,20. In contrast, the formation of vasculature inside synthetic materials requires an additional fabrication step to remove embedded sacrificial templates through pullout17,21, dissolution3,6,22,23,24,25, melting26,27,28,29, or thermal vaporization16,30,31,32,33,34,35 techniques. Vaporization of sacrificial components (VaSC) is the only method that enables the fabrication of complex vascular networks in load-bearing polymers and composites processed at high temperatures and pressures16,30,31,33. However, the VaSC process requires a sustained external heat source first to cure the host matrix and subsequently to depolymerize the sacrificial template (ca. 200 °C for 12–24 h under vacuum)31, consuming megajoules to gigajoules of external energy36. New strategies for faster matrix curing and template depolymerization at lower temperatures are thus desirable to reduce the time, energy, and complexity of fabricating vascular structures for aerospace, automotive, marine, biomedical, and renewable energy industries.

## Results and discussion

### Material system for tandem polymerization and vascularization

Inspired by the coordinated vascularization process in biological structures, we have developed a coupled polymerization and depolymerization strategy for rapid manufacturing of vascular thermosets and composites under ambient conditions (Fig. 1a). This synchronized process relies on the exothermic frontal polymerization (FP)37,38 of the dicyclopentadiene (DCPD) monomer. Here, a sacrificial template is first embedded in liquid or gelled DCPD resin. Brief (seconds), localized heating activates a latent second-generation Grubbs’ catalyst (GC2) to initiate the self-propagating reaction wave that converts a liquid or gelled resin into rigid poly(dicyclopentadiene) (pDCPD) thermoset (Fig. 1b). The surplus of energy released during the curing reaction is harnessed for concurrent depolymerization of an embedded sacrificial template. This tandem approach eliminates the need for a sustained external heat source to manufacture vascular structures, enabling efficient fabrication at room temperature (RT).

Successful coordination between polymerization and vascularization requires a sacrificial polymer that remains stable during melt processing and integration into the host matrix, yet depolymerizes efficiently and rapidly under the thermal conditions generated during FP. In situ temperature measurements during FP of DCPD resin incubated to different degrees of precure (α0) show maximum temperatures exceeding 200 °C for α0 ≤ 0.08, followed by cooling to below 100 °C within 2 min (Supplementary Fig. 1c). The chemical enthalpy released during FP decreases with increasing α0 (Eq. 1 in the “Methods” section), thus reducing the maximum temperature. Thermal vaporization of poly(lactic acid) (PLA) templates used in prior vascularization methods requires several hours at such temperatures30,31, making PLA an unsuitable sacrificial polymer for our strategy. We found that these seemingly incompatible requirements for high-temperature processing and low-temperature depolymerization are satisfied by poly(propylene carbonate) (PPC)39. In its thermally stable state, PPC is templated into fibers, sheets, and printed architectures. An orthogonal stimulus such as UV light subsequently transforms the template into a thermally unstable state for facile depolymerization.

PPC is thermally stable up to 250 °C but undergoes acid-catalyzed depolymerization into propylene carbonate (PC) monomer (Fig. 1c) at much lower temperatures (100–200 °C)39. Melt extrusion of PPC blended with an acid catalyst is not feasible due to depolymerization at typical processing temperatures (150–200 °C). Instead, the high thermal stability of a diaryliodonium-based photoacid generator (PAG) (up to ca. 180 °C39) enables the melt-spinning of PPC fibers (Mw ~ 196 kDa) blended with PAG at 155 °C without significant depolymerization (Supplementary Fig. 2a). The depolymerization onset temperature (Td, defined by 5% mass loss) of as-spun PPC (1% PAG) fibers occurs at 200 °C with complete mass loss at 230 °C (Supplementary Fig. 2b). Activation of the PAG via UV irradiation reduces Td to ca. 93 °C with complete mass loss at 130 °C, which is ca. 100 °C lower than as-spun fibers. These mass loss results suggest that UV-irradiated PPC (1% PAG) templates are suitable for depolymerization during FP of the host matrix.

### Successful vascularization window in neat thermosets

Initially, we investigated the depolymerization of a single UV-irradiated PPC (1% PAG) fiber in tandem with FP of a DCPD gel (α0 = 0.25) (Fig. 2a). The sacrificial fiber is embedded in liquid resin which cures slowly at RT into a gel with the desired α0 (Supplementary Table 1). FP is triggered by briefly powering a resistive wire oriented perpendicular to the fiber. The polymerization front propagates through the specimen at ca. 0.6 mm s−1 and vascularization trails by ca. 1.5 mm (Fig. 2a, Supplementary Video 1). A slight distortion in the shape of the polymerization front also occurs near the sacrificial fiber. Temporal delays associated with heat conduction from the matrix to the template and its subsequent endothermic depolymerization into liquid and gaseous molecules are responsible for these observations. Optical microscopy of the microchannel cross section and X-ray computed microtomography (μCT) reconstruction of the microchannel volume reveal that the dimensions and circularity of the channel replicate the sacrificial fiber (Supplementary Fig. 4 and Supplementary Table 5). As expected, as-spun fibers without UV irradiation do not depolymerize in similar experiments (Supplementary Fig. 5). Vascularization of pDCPD specimens with UV-irradiated sacrificial fibers occurs in less than 2 min under ambient conditions after triggering FP. Activation of the PAG and FP initiation require minimal energy input, ca. 108 J, reducing the thermal energy consumption by six orders of magnitude compared to ca. 120 MJ estimated for conventional curing and VaSC steps for samples of similar size (Supplementary Table 2).

The depolymerization kinetics of PPC is highly temperature-sensitive (Supplementary Fig. 6a, b) and successful vascularization depends on the surplus heat released by the host matrix during FP. Using the single sacrificial fiber geometry inside a glass mold, we explore the potential range of α0 for vascularization of the host matrix. As α0 increases, both the front velocity (Vfront) and maximum front temperature (Tfront) decrease (Fig. 2b) due to a reduction in the fraction of chemical bonds capable of ring-opening during FP38. Concurrent FP and vascularization occur for α0 ≤ 0.25, yielding fully evacuated channels. While the polymerization front still propagates in gels with 0.30 ≤ α0 ≤ 0.35, the released heat is insufficient for complete depolymerization of the sacrificial fiber, leaving partially to fully obstructed microchannels (Supplementary Fig. 8). The polymerization front does not propagate in gels with α0 ≥ 0.40 due to the reduced heat released during FP and heat loss to the glass mold.

### Numerical modeling of coupled polymerization and depolymerization

To gain insight into the tandem vascularization process, we developed a coupled thermochemical model for the thermal field (T), the degree of cure (α) of DCPD, and the degree of depolymerization (β) of PPC. The reaction-diffusion model38 was implemented in a transient, nonlinear finite-element solver (see “Methods” section). Successful curing and depolymerization are numerically defined by α ≥ 0.90 and β ≥ 0.90, respectively. The simulation shown in Fig. 2c predicts localized deceleration of the polymerization front near the sacrificial template and a spatial delay in vascularization with respect to the location of the front, which are observed experimentally in Fig. 2a. The computational results reveal that these observations are the consequence of the time required for heat to transfer from the cured pDCPD into the PPC template for endothermic depolymerization. The Vfront and Tfront predictions for different α0 also match closely with experiments in the presence of the glass molds (Supplementary Fig. 9). The model predicts successful curing and vascularization for α0 = 0.25, while also predicting unsuccessful vascularization due to incomplete depolymerization of PPC for α0 = 0.35 (Fig. 2d), as observed experimentally.

We further exploited this coupled thermochemical model to investigate the effect of a more insulating boundary condition (i.e., air) on the window for successful vascularization. Slightly higher Vfront and Tfront are predicted for an air boundary compared to a glass mold boundary (Supplementary Fig. 10). The lower convective heat loss to air compared to the conduction-driven heat loss to the glass mold allows longer heat retention in the matrix after FP, thereby extending the window for vascularization to α0 ≤ 0.35 and the window for front propagation to α0 ≤ 0.50. To compare these predictions with experiments, elastomeric DCPD gels with α0 > 0.20 containing sacrificial fibers are removed from the molds and subjected to FP. The Vfront and Tfront measurements along with the successful vascularization regime agree with the calculations from the model and the cooling profile of the matrix after FP is substantially slower (Supplementary Fig. 1c). The model successfully captures the thermochemical competition between the heat generated by the host matrix, the heat consumed by the depolymerizing fiber, and the heat lost to the surroundings, which determine the successful fabrication of vascular structures.

### Advanced manufacturing of microvascular structures

Our concurrent vascularization strategy offers excellent control over the matrix and microchannel architecture. Incubating the resin at RT to obtain viscoelastic gels proves advantageous for passive and dynamic modulation of the microchannel architecture. A range of complex vascular structures (Fig. 3a–f) is easily fabricated from elastomeric DCPD gels containing sacrificial PPC (1% PAG) templates. A 3D helical microchannel is created in pDCPD from a heat-set PPC template (Fig. 3a, b). μCT reconstruction of the microchannel reveals excellent dimensional and morphological fidelity. A flat gel with a straight fiber is bent into a horseshoe shape prior to FP without fracturing the template and this shape is preserved in the resulting microchannel (Fig. 3c). Further, this vascularization process also enables in situ modulation of the vascular geometry by applying oscillatory strain to a gel with a sacrificial template during FP (Fig. 3d and Supplementary Video 2). The polymerization front permanently sets the instantaneous out-of-plane deformation experienced by the gel and the fiber during each loading cycle (Fig. 3e). The pitch and amplitude of the channel are tuned by controlling the oscillation frequency (Fig. 3f). Such control over vascular features may find use in particle/cell separation40 or mixing of multiple fluids via non-laminar flow26.

Vascularized carbon-fiber-reinforced composites with circular microchannels are also rapidly manufactured with the FP-based method (Fig. 3g and Supplementary Table 5). The presence of the fiber reinforcement exacerbates the thermochemical competition between FP and VaSC processes due to the decreased resin volume, resulting in lower front temperatures38. Additional PAG is added to the PPC fibers to accelerate the depolymerization kinetics and achieve complete mass loss at lower temperatures (Supplementary Fig. 2c). PPC (3% PAG) fibers are sandwiched between layers of carbon-fiber fabric, then the fabric layup is infused with a liquid DCPD monomer. Briefly powering a resistive heating mat underneath the impregnated fabric stack triggers FP through the thickness of the laminate and facilitates longer heat retention in the cured composite, which ensures complete depolymerization of the sacrificial fiber. The fabrication time and energy are reduced by three and four orders of magnitude, respectively, compared to conventional curing and vascularization of similar size samples in an oven (Supplementary Table 2). We anticipate even more significant energy savings for larger components. For traditional autoclave cure and vaporization of templates at high temperature, energy consumption scales directly with part size36,38. In great contrast, the required energy input per unit volume for FP-mediated processing actually decreases with increasing component size38.

### Hierarchical interconnected vascular network

The ability to 3D print sacrificial templates enables the fabrication of multiscale vascular structures with this tandem FP and vascularization strategy. A printed sacrificial PPC (1% PAG) template resembling the primary and secondary veins of an Impatiens leaf (Fig. 4a, b) is embedded in a DCPD gel (α0 = 0.25). During FP, the liquid and gaseous depolymerization products escape through the primary vein, forming interconnected microchannels behind the polymerization front (Fig. 4c). A vascular pDCPD structure with a 3× difference in width between the central and branched veins of the hollow network is formed in 80 s under ambient conditions (Supplementary Video 3). Optical micrograph and μCT reconstruction of the specimen in Fig. 4d reveal excellent replication of the hierarchical template.

Scaling of the vascularization process for larger structures and fabrication of more complex networks will require addressing several challenges. Currently, the architectural complexity is limited by the resolution and capability of the 3D printing process for the sacrificial templates. Our experiments and modeling reveal that the total heat available for depolymerization is dictated by the enthalpy of FP, the thermal conductance at the boundaries, the surrounding temperature, and the placement of the sacrificial templates. Templates that are placed near the surfaces of the host thermoset or composite will experience faster heat loss to the surroundings compared to the center, which could impede the formation of clear microchannels. Monomers with higher enthalpy of FP can increase the heat available for depolymerization. Alternatively, replacing metallic tooling surfaces with more thermally insulating materials such as rigid polyisocyanurate foams will prolong heat retention in large structures and aid complete depolymerization. The maximum size of a template and the irradiation time are dictated by the dimensions and the intensity of the UV lamp, respectively. We envision a narrowband LED source with a higher intensity for continuous irradiation of the fiber during spooling to circumvent these limitations. The maximum thickness of the templates that undergo complete depolymerization depends on sufficient activation of the PAG. Short-wave UV irradiation may not penetrate well through templates thicker than 600 μm, leading to an insufficient amount of activated PAG. Adding a photosensitizer to extend the absorption window to longer wavelengths will be beneficial in overcoming this challenge. The experimental methods and computational tools that we have developed can be used to investigate these different design parameters and help build a library of compatible host resins and sacrificial components for synchronized fabrication of large-scale vascular structures.

We have successfully exploited the frontal polymerization of a structural host matrix in tandem with the depolymerization of a sacrificial template to create a range of vascular structures in a rapid, energy-efficient single-step process. Excellent control over the microchannel shape, size, and complexity was achieved with orthogonally triggered sacrificial templates. Vascularization of thermosets with interconnected microchannels was achieved within a few minutes under ambient conditions with energy savings of six orders of magnitude over competing techniques. Our thermochemical model successfully captures the competition between the reaction enthalpy of the host matrix, the energy consumed during template depolymerization, and the energy dissipated into the surroundings. The model provides a promising tool for the design of host matrix and sacrificial components for FP-based synchronized manufacturing of vascular structures. Tandem polymerization and vascularization will propel the use of high-performance vascular structures by reducing the energetic and temporal footprint for fabrication by several orders of magnitude.

## Methods

### Materials

Dicyclopentadiene (DCPD), 5-ethylidene-2-norbornene (ENB), second-generation Grubbs’ catalyst (GC2), phenylcyclohexane (PCH), tributyl phosphite inhibitor (TBP), ethanol, and 5-(N-2,3-Dihydroxypropylacetamido)-2,4,6-triiodo-N,N′-bis(2,3-dihydroxypropyl)isophthalamide (HistodenzTM) are purchased from Sigma-Aldrich and used as received. Poly(propylene carbonate) (PPC) pellets are acquired from Novomer. 4-Methylphenyl[4-(1-methylethyl)phenyl]iodonium tetrakis(pentafluorophenyl)borate photoacid generator (PAG), commercially known as Rhodorsil-FABA, is gifted by Bluestar Silicones. The reinforcement for vascular composite specimens is Toray T300 carbon fiber 2 × 2 twill weave fabric. A 26-gauge Kanthal wire is used to trigger FP of neat resin specimens in glass molds. Post-FP, steel wires are used to verify channel clearance in neat and FRPC specimens.

### Resin formulation for frontal polymerization

Since DCPD is solid at 20 °C, we melt it in an oven at 40 °C and then add 5 wt.% ENB to depress the freezing point below room temperature (RT). For a typical experiment, the FP-capable resin is prepared by mixing 62.7 mg GC2 with 3.12 mL of PCH using a sonication bath. TBP (20 μL, i.e., one molar equivalent with respect to GC2) is added to the catalyst solution using a volumetric syringe. The GC2/PCH/TBP solution is then thoroughly mixed with 97.4 g DCPD/ENB solution (10,000 molar equivalents with respect to GC2). The resulting resin is used for experiments involving a neat matrix. For composite experiments, 0.3 molar equivalents of TBP are used. Unless otherwise specified, all FP experiments are performed under standard ambient conditions.

### Front temperature and velocity measurements

For both neat matrix and composite experiments, a T-type thermocouple (TMQSS, Omega) is inserted into the specimen midplane to measure the peak front temperature. The thermocouple is placed far away (ca. 15 mm) from embedded sacrificial fibers to avoid any influence on the polymerization front in the vicinity of the sacrificial fiber. The thermocouple is also far away from the resistive heating wire for temperature measurements during steady-state propagation of FP.

For the neat matrix experiments, a Canon EOS 7D digital camera with a 100 mm macro lens is focused on the plane of the sacrificial fiber embedded in the specimen. Tracker software (Open Source Physics) is used to determine the location of the polymerization front along the specimen length based on color and refractive index mismatch between the uncured and cured matrix in the optical images extracted from the Canon video. The front velocity is determined from the slope of the best-fit trendline for the position of the polymerization front as a function of time.

### Resin rheology

The storage and loss moduli of the DCPD resin as it incubates (slowly cures) under isothermal conditions (20 °C) are extracted from time sweep measurements at 0.1% strain and 1 Hz frequency on an AR-G2 rheometer (TA Instruments). Approximately 0.5 mL of resin is placed between a 25-mm-diameter steel top plate equipped with a solvent trap and a bottom Peltier plate on the instrument for each test. A loading gap of ca. 1000 μm is used for each test. A storage modulus (G′) > 30 kPa indicates the formation of elastomeric gels (degree of precure, α0 > 0.20), which can be removed from glass molds without any tack on the contact surfaces (Supplementary Fig. 1b). Gels with G' < 30 kPa (α0 < 0.20) are difficult to demold cleanly due to deformation of the viscous resin, resulting in irregular surface defects on the cured specimen post-FP.

### Thermal characterization of resin

The degree of precure (α0) of the resin is tracked during incubation at 20 °C using differential scanning calorimetry (DSC) in a Discovery DSC Q250 (TA Instruments) equipped with an RCS 90 cooling system. Approximately 2–3 mg of resin is placed in an aluminum hermetic DSC pan and sealed before loading onto the DSC autosampler. Loading a small mass of each sample ensures a constant temperature gradient in the instrument despite the highly exothermic nature of the curing reaction. Each sample is subjected to a constant ramp rate of 7 °C min−1 between −50 and 250 °C to determine the thermal cure profile (Supplementary Fig. 1a). The baseline-corrected exothermic peak in the heat flow signal is integrated to extract the enthalpy of reaction (Hr1). The degree of precure of each sample is then calculated using Eq. 1:

$${\alpha }_{0}=1-\frac{{H}_{{\rm{r}}1}}{{H}_{{\rm{r}}0}}$$
(1)

where Hr0 is the enthalpy of reaction of freshly prepared liquid (uncured) resin and Hr1 is the remaining enthalpy of gelled (partially cured) resin. Degree of precure (α0) of the DCPD resin for different incubation times is summarized in Supplementary Table 1. The specific heat capacity of the matrix (Cp1) is averaged between 20 and 200 °C after calibration with a sapphire standard.

### Melt processing of sacrificial templates

Melt-blending of PPC with PAG is performed below 160 °C to minimize acidic depolymerization of PPC through thermal activation of the PAG during processing. Neat PPC pellets are blended with 1–3% PAG in a twin-screw melt compounder (Type Six, C.W. Brabender® Instruments). The twin-screw chamber is heated to 140 °C after which 40 g of PPC pellets are added to the chamber while rotating the mixing blades at 15 RPM until completely melted (ca. 10 min). The desired amount of PAG powder (per wt. % of PPC) is slowly added and then mixed at 45 RPM for 5 min. The melt-compounded material is removed from the chamber, cut into small pieces while still soft, then allowed to cool to RT.

Sacrificial fibers and filaments are produced using a modified lab-scale melt-spinning apparatus (extruder). The blended polymer (20 g) is fed into a steel barrel and melted at 155 °C for 10 min. A polytetrafluoroethylene (PTFE) disc followed by a brass-capped steel piston is inserted into the top opening of the barrel to prevent the melted polymer from sticking to the piston. The sacrificial polymer is extruded through a 1.25 mm diameter spinneret at 3 g min−1 by mechanically advancing the piston. A metal pulley translates laterally and guides the extruded fiber to wrap uniformly around a spool mounted on an 88-mm diameter winding reel. The surface speed of the take reel is adjusted to control the final diameter of the fiber. For example, a surface speed of 120 m min−1 produces roughly 200 m of 400 μm diameter fiber. The sacrificial filaments are extruded through the same apparatus under the same temperature settings by replacing the smaller spinneret with a 2.5-mm diameter spinneret connected to a 75-mm-long hollow brass extension. The extruded filament (ca. 2.85 mm diameter) enters a 1 m tall water column at RT instead of winding on a take-up drum. The filaments are vacuum dried at 40 °C for 24 h before feeding into the 3D printer.

An Impatiens leaf is scanned and its digital image is imported into ImageJ for mapping the key coordinates of the primary and secondary veins (Supplementary Fig. 12). The coordinates are then converted into a g-code and loaded into a TAZ6 fused deposition modeling printer (Lulzbot). The spun PPC (1% PAG) printing filament is fed into the printer and extruded at 160 °C through a 500 μm diameter nozzle onto an 80 °C bed. The template is printed at a 0.4 mm print height and 35 mm min−1 material feed rate.

### Photoactivation of acid catalyst

Activation of the PAG is carried out on an IntelliRay UV0320 parabolic flood curing system (Unitron International). Sacrificial fibers (15 cm long and 400–600 µm diameter) or printed templates are placed on an aluminum foil and taped at the ends to prevent shrinking during irradiation. The foil is secured on an adjustable metal shelf 75 mm below the lamp such that all the fibers are within the area projected by the lamp. The fibers are irradiated for a total of 10 min at 100% power (6–8 J cm−2 at 280 nm). The templates are flipped after 5 min to promote activation of the PAG through the thickness of the sacrificial polymer. The templates are either used immediately after irradiation or stored at −20 °C in a lightproof box until use. The UV irradiation time for the templates is not optimized and a more powerful UV source would be ideal for activating the PAG rapidly within seconds. Unless otherwise specified, all references to the sacrificial fibers or templates refer to UV-irradiated material.

### Characterization of sacrificial templates

Scanning electron micrographs (SEM) of the fibers are acquired on a FEI Quanta FEG 450 Environmental SEM. Samples are imaged at 5 kV after sputter coating with gold/palladium for 70 s (ca. 7 nm) using a Denton Desk II TSC-turbo-pumped sputter coater.

Molecular weights of pellets and fibers are obtained via analytical gel permeation chromatography (GPC) on a system composed of a Waters (1515) Isocratic high-pressure liquid chromatography pump, a Waters (2414) Refractive Index Detector, a Waters (2707) 96-well autosampler, and a series of 4 Waters HR Styragel columns (7.8 × 300 mm, HR1, HR3, HR4, and HR5) in tetrahydrofuran (THF) at 30 °C. The GPC is calibrated using monodisperse polystyrene standards. Each sample (10 mg) is weighed on an analytical balance (XPE205, Mettler-Toledo) and dissolved in 1 mL THF. The polymer solution is injected into a 300 μL GPC vial through a 0.45 μm polytetrafluoroethylene (PTFE) filter to remove any undissolved particulates before loading it into the GPC autosampler. Number- and weight-averaged molecular weights (Mn and Mw) are calculated using Wyatt’s Astra 6 software through the integration of the signal peaks obtained from the RI detector (Supplementary Fig. 2a).

Thermal depolymerization via mass loss of sacrificial templates (3 mg samples) in a nitrogen environment was measured on a TGA (Q500, TA Instruments) equipped with an evolved gas analysis furnace and calibrated with nickel standards (Supplementary Fig. 2b, c). For dynamic tests, the mass loss is recorded during a heating cycle over the temperature range from 40–250 °C at a linear ramp rate of 1–100 °C min−1. Instrument limitations caused a nonlinear increase in initial temperature from 40 to 100 °C for high ramp rates (>40 °C min−1). For isothermal tests, the temperature is ramped from 40 to 10 °C below the desired temperature at a linear ramp rate of 10 °C min−1, then subsequently ramped to the desired temperature at a slower linear rate of 5 °C min−1 to minimize temperature overshoot. The isothermal temperature is maintained for 3–6 h for each test.

The glass transition temperature (Tg), heat capacity (Cp2), and enthalpy of depolymerization (Hr2) of PPC fibers are measured on the DSC Q250 (Supplementary Fig. 6c, d). Approximately 2–3 mg of the sacrificial polymer is sealed in an aluminum hermetic DSC pan and exposed to a constant ramp rate of 5 °C min−1 between −50 and 180 °C. The specific heat capacity of the fibers is averaged between 20 and 140 °C after calibration with a sapphire standard. The Hr2 of the sacrificial polymer is extracted by integrating the baseline-corrected endothermic peak in the heat flow signal between 50 and 180 °C. For open-pan DSC experiments, a TA instrument Q20 DSC connected to a CFL-50 cooling system is used. The enthalpy of vaporization of PC monomer (integrated between 60 and 150 °C) is subtracted from the enthalpy of vaporization of PPC (1% PAG) fiber to estimate the enthalpy of depolymerization of the fiber.

### Specimen fabrication for vascularized neat polymer

For FP of specimens in glass molds, a single PPC (1% PAG) fiber is clamped between two polyurethane spacers (inner dimensions: 60.0 mm × 34.0 mm × 6.0 mm) and sandwiched between two 76.2 mm × 50.8 mm × 6.4 mm glass plates in a cell casting mold (Supplementary Fig. 3). Liquid resin is poured into each mold to completely submerge the fiber. A resistive wire perpendicular to the fiber orientation is then placed in the resin. Resin-filled molds are incubated in an environmental test chamber (MicroClimate, Cincinnati Sub-Zero Products) at 20 °C to achieve the desired degree of precure (α0) (Supplementary Table 1). Once the desired α0 is reached, FP of the gelled sample is initiated by briefly powering (6 W for ca. 3 s) the resistive wire.

For FP of specimens without the glass mold, the precured elastomeric gel specimens are removed from the cell casting mold once α0 > 0.20. The edges of the specimens are trimmed with a razor blade to expose the transverse ends of the sacrificial fiber. FP is then initiated by applying a soldering iron to one end of the specimen.

For the helical microchannel in Fig. 3a, b, a PPC fiber (1% PAG) is heat-set at 40 °C into the desired shape before embedding in the glass mold and filling with resin.

For dynamic modulation experiments in Fig. 3d–f, DCPD gels containing a single sacrificial fiber (located at 1 mm from the surface of a 4-mm-thick sample) are fixed between tensile grips on an Instron 8841 fatigue frame with a gauge length of 50 mm. FP is thermally initiated using a soldering iron at one end of the sample, and dynamic (sinusoidal) loading is applied as the front propagates. Front propagation, applied tension, and fiber orientation are all in the same direction. The displacement amplitude is maintained at 4 mm for all samples and the frequency is varied from 0.1 to 1 Hz.

For the leaf specimen in Fig. 4, the PPC (1% PAG) template was submerged in liquid DCPD resin inside a glass mold that was subsequently incubated at RT for ~12 h prior to initiating FP. The gelled specimen (α0 ~ 0.25) was removed from the mold and a soldering iron was used to trigger the FP on one side perpendicular to the primary vein of the template.

### Microchannel characterization and reconstruction

The vasculature is rinsed with ethanol after FP and optical images of polished cross sections are captured on a Keyence VHX-5000 digital microscope at 100×-200× magnification. Microchannels within the host matrices are also visualized with X-ray computed microtomographic (μCT) imaging (Xradia MicroXCT-400 using TXM Controller software). The μCT scans are conducted after infiltrating the empty microchannels with a contrast solution (350 mg HistodenzTM in 1 g DI water). For 5-mm long channels, 360° scans are obtained in rotation intervals of 0.4° with a ×4 objective (5 μm per pixel) at 5 s exposure times with 40 kV (200 μA, 8 W) source. For 20-mm long microchannels, imaging settings are changed to rotation intervals of 0.5° with a ×1 objective (20 μm per pixel) at 3 s exposure times with 60 kV (133 μA, 8 W) source. Scan reconstructions are completed in a proprietary software and images are produced via AmiraTM (v. 6.4.0). ImageJ software is used to measure the dimensions and circularity of the channels from optical images and μCT reconstructions (Supplementary Table 5).

### Computational modeling of coupled polymerization and vascularization

The frontal polymerization of DCPD and the depolymerization of PPC are coupled thermochemical processes described by their own reaction-diffusion model. For computational efficiency, we simulate the PPC fiber embedded in a cylindrical DCPD domain (Supplementary Fig. 7a) in an axisymmetric setting. In the DCPD domain, the thermochemical model expressed in terms of the temperature, T (in K), and degree-of-cure, $$\alpha$$, takes the form

$$\left\{\begin{array}{c}{\hskip-1pc\kappa }_{1}\frac{{\partial }^{2}T}{\partial {r}^{2}}+{\kappa }_{1}\frac{1}{r}\frac{\partial T}{\partial r}+{\kappa }_{1}\frac{{\partial }^{2}T}{\partial {z}^{2}}+{\rho }_{1}{H}_{{\mathrm{r}}1}\frac{\partial \alpha }{\partial t}={\rho }_{1}{C}_{{\mathrm{p}}1}\frac{\partial T}{\partial t}\\ \frac{\partial \alpha }{\partial t}={A}_{1}\exp \big(-\frac{{E}_{a1}}{\mathrm{RT}}\big){(1-\alpha )}^{{n}_{1}}{\alpha }^{{m}_{1}}\frac{1}{1+\exp [C(\alpha -{\alpha }_{\mathrm{c}}-{\alpha }_{0})]}\end{array}\right.$$
(2)

where the subscript “1” denotes quantities associated with DCPD, α = 0 denotes the uncured resin, and α = 1 denotes the fully cured polymer. In (2), κ (in W m−2 K−1) denotes the thermal conductivity, ρ (in kg m−3) the density, Cp (in J g−1 K−1) the specific heat, and Hr (in J g−1) the total enthalpy of reaction. The second equation in (2) captures the cure kinetics, with A (in s−1) denoting the pre-exponential constant, Ea (in kJ mol−1) the activation energy, and R (= 8.314 J mol−1 K−1) the universal gas constant. The additional terms appearing in the cure kinetic model are based on the Prout–Tompkins auto-catalytic model41 $${(1+\exp [C(\alpha -{\alpha }_{\mathrm{c}}-{\alpha }_{0})])}^{-1}$$, which also accounts for α0. This model has been successfully used to simulate FP in DCPD monomer42 and DCPD-based composites43,44,45. Supplementary Table 3 provides the material properties and Supplementary Table 4 provides the cure kinetic parameters for DCPD.

The depolymerization process of the PPC fiber is described similarly in terms of the thermal field and the degree of depolymerization $$\beta$$, with $$\beta =0$$ corresponding to the intact PPC and $$\beta =1$$ the fully depolymerized state:

$$\left\{\begin{array}{c}{\kappa }_{2}\frac{{\partial }^{2}T}{\partial {r}^{2}}+{\kappa }_{2}\frac{1}{r}\frac{\partial T}{\partial r}+{\kappa }_{2}\frac{{\partial }^{2}T}{\partial {z}^{2}}-{\rho }_{2}{H}_{{\mathrm{r}}2}\frac{\partial \beta }{\partial t}={\rho }_{2}{C}_{{\mathrm{p}}2}\frac{\partial T}{\partial t}\\ \hskip-4pc \frac{\partial \beta }{\partial t}={A}_{2}\exp\, \big(-\frac{{E}_{{\mathrm{a}}2}}{\mathrm{RT}}\big){(1-\beta )}^{{n}_{2}}{\beta }^{{m}_{2}}\end{array}\right.$$
(3)

where the subscript “2” denotes quantities associated with PPC, and Hr2 the total heat absorbed per unit mass by the depolymerization of the sacrificial material. To extract the parameters defining the depolymerization kinetics of PPC, a constrained nonlinear multi-variable optimization algorithm46 is adopted to match dynamic TGA data obtained at different ramping rates (Supplementary Fig. 6a, b). The material properties and depolymerization kinetics parameters used in this study for PPC are listed in Supplementary Tables 3 and 4, respectively.

As shown in Supplementary Fig. 7b, convective boundary conditions (with a film coefficient h and an ambient temperature of 20 °C) are applied along the outer surfaces of the cylindrical domain. In the simulation, the DCPD radius (r1) is 3 mm, the fiber radius (r2) is 200 μm, and the length of the domain (l) is 20 mm. Initiation of the FP process is achieved by prescribing a temperature (200 °C) for one second along the edge at x = 0. The film coefficient is chosen (h = 25 W m−2 K−1) such that the temperature profile in the DCPD is similar to that measured in the experiments. For the simulations involving a glass mold, a glass layer is added along the top of the simulation domain with a glass thickness of 0.7 mm to capture the temperature history (Supplementary Fig. 7c). In that glass layer, the transient heat conduction equation (in the absence of the reaction term) is solved, with the convective boundary condition applied to the surface of the glass layer exposed to air.

The analysis is performed with Multiphysics Object-Oriented Simulation Environment (MOOSE), an open-source C++ math library and nonlinear finite-element solver47 that provides the mesh adaptivity needed to capture accurately and efficiently the sharp gradients in temperature and degree of cure/depolymerization in the vicinity of the advancing front.

### Specimen fabrication for vascularized fiber-reinforced composites

Composites specimens are fabricated with 10 plies of 2 × 2 twill weave carbon-fiber fabric. The DCPD resin solution is prepared with 0.3 equivalents of TBP with respect to GC2. A wet layup technique is used to infuse the liquid DCPD resin into a 10 cm by 13 cm stack of dry carbon-fiber fabric with a PPC fiber (3% PAG, 400 μm diameter) secured between the fifth and sixth layer (midplane of the fabric stack) (Supplementary Fig. 11). The layup is prepared on a resistive heater (OMEGALUX®), which is secured onto an insulator tool plate (448-D, Fibre Glast Developments Corp.). Silicone spacers (4-mm thick) are placed adjacent to the layup to dictate the cured composite thickness. An additional insulator tool plate is rested on top of the layup, then the entire setup is moved to a hydraulic press (MTP-13, Tetrahedron). A platen force of 2 kN is applied onto the setup and then FP is initiated by powering the resistive heater. The heater was switched off as soon as a spike in the thermocouple temperature was observed (ca. 30 s) indicating that the front has propagated through the composite thickness (Supplementary Fig. 1d).

Depolymerization of the sacrificial fiber post-FP is evaluated by passing a steel wire through the channel. The channel fidelity and void content in the cured composite are measured by polishing the cross sections of samples cut along the panel, then imaging the polished surfaces via a digital optical microscope (VHX-5000, Keyence). ImageJ software is used to measure the circularity of the channel and to calculate the ratio of void to cross-sectional area for each polished sample. The composite fiber volume fraction, Vf, is calculated using Eq. 4:

$${V}_{{\rm{f}}}=\frac{{f}_{{\rm{A}}}n}{{\rho }_{{\rm{f}}}z}$$
(4)

where fA is the areal weight of the fabric, ρf is the fiber density, n is the number of plies, and z is the composite thickness.

### Energy and time consumption calculations

The energy consumed during any heating step is determined by multiplying the energy consumption rate by the step time. The energy consumption rate during ramp and dwell steps for an oven with a power rating of 4.8 kW and an interior volume of 0.8 m3 were 0.046 and 0.024 kWh min−1, respectively36. The following protocols were used for each process:

BPA-epoxy oven-cure cycle: ramp from 20 to 121 °C (50 min), dwell at 121 °C (2 h), ramp to 177 °C (25 min), and dwell at 177 °C (3 h).

DCPD oven-cure cycle: ramp from 20 to 70 °C (30 min), dwell at 70 °C (2 h), ramp to 170 °C (50 min), and dwell at 170 °C (1.5 h).

PLA VaSC cycle: ramp from 20 °C to 200 °C (2 h) and dwell at 200 °C (12 h).

DCPD FP initiation with resistive wire: A DC source with 3 A at 2 V was switched on for 3 s.

FRPC FP initiation with heating mat: A 300 W heater was powered on for 30 s.

UV irradiation cycle for PPC: 10 min at 10 mW cm−2 for a 15 cm long fiber.

## Data availability

Source data are provided with this paper. Any supplementary data that support the findings of this study are available from the corresponding author on reasonable request.

## Code availability

We performed the simulations by developing an application based on MOOSE, an open-source C++ finite-element framework developed at Idaho National Laboratory. MOOSE is freely available at mooseframework.org. A compiled version of the code for simulating the synchronized manufacturing process, an example input file, and the instructions for running the simulations are provided and can be accessed under the MIT License at https://gitlab.com/zhangx22/concurrentvascfp. These files could be used to reproduce the computational results reported in this manuscript or simulate the tandem polymerization and depolymerization processes with a different geometrical settings, initial or boundary conditions.

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## Acknowledgements

We acknowledge funding from the Air Force Office of Scientific Research (AFOSR #FA9550-20-1-0194, Center of Excellence in Self-healing and Morphogenic Manufacturing). M.G. was supported by a PPG-MRL Graduate Fellowship, L.M.D. was supported by an NSF Graduate Fellowship, E.M.L. was supported by a Beckman Foundation Graduate Fellowship. We thank J.L. Vazquez for their graphics assistance. We also acknowledge the late Prof. S. R. White who helped conceive this idea and contributed to the research.

## Author information

Authors

### Contributions

N.R.S., J.S.M., and P.H.G. directed the research. M.G. and I.D.R. conceived the idea. M.G., J.E.A., P.J.C., L.M.D., E.M.L., and M.Y designed the experiments and X.Z. conducted the computational studies. For the experimental work, M.G. focused on sacrificial polymer characterization and vascularization in neat thermosets, J.E.A. performed the 3D printing and rheological characterization, P.J.C. manufactured the vascular composites, L.M.D. helped with the oscillatory tests, and Y.L. performed the front velocity measurements. All authors participated in writing the manuscript.

### Corresponding author

Correspondence to Nancy R. Sottos.

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### Competing interests

The authors have filed patent application US16809769 related to this work.

Peer review information Nature Communications thanks Edward Archer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Garg, M., Aw, J.E., Zhang, X. et al. Rapid synchronized fabrication of vascularized thermosets and composites. Nat Commun 12, 2836 (2021). https://doi.org/10.1038/s41467-021-23054-7

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