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15 February 2021, Volume 42 Issue 02
• Invited Paper
Promotion mechanism of color fastness to sublimation in thermovacuum environmental conditions for fibroin powder/pigment complex
CAO Genyang, WANG Yunli, SHENG Dan, PAN Heng, XU Weilin
Journal of Textile Research. 2021, 42(02):  1-6.  doi:10.13475/j.fzxb.20210106206
Abstract ( 301 )   HTML ( 13 )   PDF (6065KB) ( 206 )
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In order to improve the thermal sublimation fastness of pigment particles on colored fabrics under thermovacuum environmental conditions, surface morphology, particle size and molecular structure of naphthol red F3RK pigment particles and silk fibroin powders were investigated, and the color of the fabric was tuned by mixing pigment particles with silk fibroin powder at different ratios. The subatomospheric pressure is used to simulate the thermal force field in the thermalvacuum environment, under which the adsorption performance was studied under different mixing ratios between pigment particles and silk fibroin powder. The interaction between silk fibroin powders and pigment particles were discussed, and the theoretical model was established. The results show that the silk fibroin powder is effective in improving thermal sublimation of colored fabrics in thermovacuum environmental conditions due to the synergistic effect of its large specific surface area and amido bonds. Employment of this method increases thermal sublimation fastness to level 5. The results of this study are of referencing significance for improving the thermal sublimation fastness of colored fabrics under extreme environmental conditions.

Fiber Materials
Preparation, structure and properties of high-strength high-modulus polyimide fibers containing benzimidazole moiety
ZHENG Sensen, GUO Tao, DONG Jie, WANG Shihua, ZHANG Qinghua
Journal of Textile Research. 2021, 42(02):  7-11.  doi:10.13475/j.fzxb.20200708006
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In order to improve the mechanical properties of polyimide (PI) fibers and promote their applications for composites, the spinning dopes were synthesized by an aromatic heterocyclic diamine monomer of 2-(4-aminophenyl)-5-aminobenzimidazole (BIA) with rigid polyimide backbones of 3,3',4,4'-biphenyl tetracarboxylic dianhydride (BPDA) and p-phenylene diamine (p-PDA), which were based on a structural design of high-performance polymeric fibers. The PI fibers were prepared via dry-spinning the BPDA-PDA-BIA solution. The relationships between mechanical properties of the fibers and chemical construction and aggregation structure were studied. In addition, the thermal properties and ultraviolet (UV) irradiation resistance of the fibers were evaluated systematically. The results show that the PI fibers reach the optimum tensile strength and initial modulus of 4.04 and 130 GPa, respectively, mainly due to the high orientation of the polymer chains along the fiber direction and the intermolecular hydrogen bonding interactions. The glass-transition and 10%-weight-loss temperature of the PI-4 fiber are 324 and 587 ℃, and the fiber retains 92% of its tensile strength after 168 h of UV irradiation, indicating prominent UV irradiation resistance and excellent thermal stability.

Preparation and optical properties of flexible photonic crystal film for structural colors
WANG Xiaohui, LI Yichen, LIU Guojin, TANG Zuping, ZHOU Lan, SHAO Jianzhong
Journal of Textile Research. 2021, 42(02):  12-20.  doi:10.13475/j.fzxb.20201007209
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To solve the problem that photonic crystal (PC) structures constructed by hard nanospheres are easy to be damaged by external forces, resulting in poor durability of the structural colors, polystyrene/poly (methyl methacrylate-butyl acrylate) (PS/P(MMA-BA)) nanospheres were synthesized by means of the step-by-step polymerization method, and the flexible PC films with structural colors were fabricated by using the prepared PS/P(MMA-BA) nanospheres as assembly blocks through melt-shear induced self-assembly method. The results show that the core cross-linking agent, divinylbenzene (DVB), is able to increase the cross-linking density and refractive index of PS core, to produce a certain refractive index contrast with the outer layer, and to make the PS core surface have double bonds, covalently bonding with the outer layer of P(MMA-BA). The outer layer cross-linking agent, allyl methacrylate (ALMA), improves the stability of the outer layer by forming some crossing-linking bonds. When DVB is 12.5% of styrene (St) and ALMA is 0.3% of the total amount of MMA and BA, PC film demonstrates excellent optical and mechanical properties. PC structures assembled by the nanospheres exhibit excellent flexibility, brilliant structural colors, and high stability during the bending, washing and rubbing tests. The melt-shear assembly method adapted to the PS/P(MMA-BA) nanospheres can rapidly fabricate large area of flexible PC film with structural colors.

Preparation of nano-tungsten oxide composite cotton fiber and its photochromic properties
WANG Yuting, LING Zhongwen, YANG Xin, LIU Yuqing
Journal of Textile Research. 2021, 42(02):  21-26.  doi:10.13475/j.fzxb.20201006007
Abstract ( 184 )   HTML ( 5 )   PDF (5408KB) ( 65 )
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In order to solve the problems of unstable crystal structure, long photochromic response time and short life of tungsten oxide, light yellow tungsten oxide nanorods were prepared by the hydrothermal one-step method for uniform dispersion into polyvinyl alcohol solution with aid of magnetic stirring. Cotton fiber was coated with tungsten oxide nanorods by rapid and continuous impregnation coating process, and its structure and properties were characterized. The results show that under the irradiation of ultraviolet light, the color of photochromic fiber changes from light yellow to grayish blue in 1 min, and gradually to dark blue in 5 min, and the color of photochromic fiber returns to its initial state in about 2 h during infrared heating treatment, showing a fast and reversible color switch from light yellow to dark blue. The photochromic fiber can be produced on a large scale and can be woven into various patterns, which has great potential in developing photochromic textiles.

Preparation and properties of continuously produced electric-responsive liquid crystal fibers
SHENG Mingfei, WANG Wanning, ZHANG Liping, FU Shaohai
Journal of Textile Research. 2021, 42(02):  27-33.  doi:10.13475/j.fzxb.20201009007
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To solve the color changing problem in flexible display made of electro-responsive liquid crystal fibers without an external electrode, this study developed a preparation process for electrochromic liquid crystal (ECLC) fibers with a coaxial sandwich structure. The ECLC fiber with conductive fiber for the inner conductive layer, conductive hydrogel for the outer transparent conductive layer, polymer dispersed liquid crystal (PDLC) as the electrochromic layer was prepared via a continuous coating method. The influence of conductive fiber types and PDLC components on the morphology and electrochromic properties of ECLC fibers was explored. The results show that when the carbon/polyamide fiber is as the core material and the mass ratio of R-CLC/polymethyl methacrylate/CH2Cl2 is set to 15∶1∶8, the prepared ECLC fiber has an obvious three-layer core-shell structure demonstrating good electrochromic performance in low driving voltage (40 V) and fast response time (9 s). The electrode-less ECLC fiber device developed in this study lays the foundation for the development of smart flexible display textiles.

Preparation and properties of polyacrylonitrile/carboxyl styrene butadiene latex composite nanofibrous membranes
GUO Xuesong, GU Jiayi, HU Jianchen, WEI Zhenzhen, ZHAO Yan
Journal of Textile Research. 2021, 42(02):  34-40.  doi:10.13475/j.fzxb.20200804607
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In order to improve the mechanical properties of electrospun polyacrylonitrile (PAN) nanofiber membranes, a series of PAN/SBR composite nanofibrous membranes were prepared by immersing PAN nanofibrous membranes into carboxyl styrene butadiene latex (SBR) solution. The effects of SBR content on the surface morphology, chemical structure, wettability, thermal and mechanical properties of PAN nanofibrous membranes were investigated. The results reveal that SBR is physically bonded to PAN nanofibers. With the increase of SBR content, the distribution of SBR on the surface of PAN fiber gradually becomes uneven. The addition of SBR has no effect on the thermal stability of PAN membrane, but increases the water contact angle and jeopardizes the hydrophilicity of the membrane. At the same time, adding a small amount of SBR (less than or equal to 15.6%) is found to increase the breaking strength, breaking elongation, initial modulus, breaking energy and puncture resistance of PAN nanofibrous membrane. When the content of SBR is 8.5% of PAN membrane, the breaking energy of the composite membrane is increased by about 4 times, which significantly improves the mechanical properties of PAN nanofiber membrane.

Protein adsorption and cell response on bio-interfaces of silk fibroin/octacalcium phosphate composites
YANG Ya, YAN Fengyi, WANG Hui, ZHANG Keqin
Journal of Textile Research. 2021, 42(02):  41-46.  doi:10.13475/j.fzxb.20201008606
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To further understand the structure-function relationships for biomimetic materials for the purpose of designing a favorable implant microenvironment for osseointegration in vivo, the nano/micro hierarchical structure of silk fibroin (SF)/octacalcium phosphate (OCP) composite coating was constructed by controlling simply the SF concentration of the electrolyte during the electrochemical deposition process. The influence of SF concentration on surface morphology, mechanical property, protein adsorption and cell proliferation were studied. The results show that the pore diameter of the coating surface decreases from (19.96±6.96) μm to (1.56±0.22) μm and the crystal width decreases to the nano-scale with SF concentration increasing in electrolyte. Comparing the pure OCP coating, the elastic modulus and hardness of the SF/OCP composite coating (1.0 mg/mL SF) increase approximately 1.5 times and 4.3 times, respectively. The SF/OCP composite coating selectively enhances fibronectin (Fn) adsorption, and the cell viability on the SF/OCP composite coating (1.0 mg/mL SF) is 1.28 times that on the pure OCP coating after culturing for 7 days.

Preparation of flax layered nano-cellulose and properties of its reinforced thermoelectric composites
HU Jing, ZHANG Kaiwei, LI Ranran, LIN Jinyou, LIU Yuqing
Journal of Textile Research. 2021, 42(02):  47-52.  doi:10.13475/j.fzxb.20201005507
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In order to prepare layered nano-cellulose from natural flax fiber, i.e. coexistence of cellulose nanofiber (CNF) and cellulose nanocrystalline (CNC), and improve the preparation method of CNF from flax fiber, a method of alkaline treatment of flax fiber in a specific concentration of sodium hydroxide solution, followed by tempo mediated ternary oxidation and mechanical treatment, was proposed. The CNF/graphene composite films were prepared by compounding flax CNF with graphene to study the properties of flax CNF reinforced thermoelectric composites. The results show that the diameter and length of flax fiber become smaller and the hemicellulose is removed after alkaline pretreatment, which is an important step towards generation of CNF with different sizes. The optical transmittance of CNF suspension increases from 3.7% to 95.1% with the increase of NaOH concentration in the range of 0-18%, and the highest power factor of CNF/graphene composite film is 8.0 × 10 -3 μW/(M·K 2), which indicates that the composite film has thermoelectric properties.

Preparation and properties of sodium alginate/krill protein scaffold materials
YIN Juhui, GUO Jing, WANG Yan, CAO Zheng, GUAN Fucheng, LIU Shuxing
Journal of Textile Research. 2021, 42(02):  53-59.  doi:10.13475/j.fzxb.20201008107
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In order to improve the strength and bio-compatibility of sodium alginate based scaffold materials made from polyelectrolyte composite system with sodium alginate (SA) and krill protein (AKP), SA/AKP scaffold materials were prepared by inducing phase separation-chemical cross-linking method at low-temperature. The structural morphology, mechanical properties, porosity, air permeability, water absorption and bio-compatibility of different concentrations of SA/AKP solutions were compared. The results show that when the concentration of SA/AKP sample is 4%, the pore size of the stent is between 20 and 96 μm with uniform and regular shape. The porosity, and absorptive characteristics of liquid and air permeability of the tissue scaffold decreased with the increasing concentration of SA/AKP solutions, and the fracture strength and elongation of the scaffold increased with the increasing concentration of SA/AKP solutions. When the concentration is 4%, the porosity of the scaffold material is 86.4%, the air permeability is 58.9%, while offering high strength and elongation. The cell toxicity grade of the scaffold is 0, indicating excellent biological safety.

Textile Engineering
Exploration of image-based testing method for yarn twist in air-jet vortex spinning
LI Hao, XING Mingjie, SUN Zhihao, WU Yao
Journal of Textile Research. 2021, 42(02):  60-64.  doi:10.13475/j.fzxb.20201008005
Abstract ( 229 )   HTML ( 1 )   PDF (3844KB) ( 65 )
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In order to find an effective method suitable for yarn twist test produced by air-jet vortex spinning, with the help of scanning electron microscope, the twist of the yarn was tested by the appearance and cross-section images of the air-jet vortex spinning yarns, based on the comparative analysis of yarns made via air-jet vortex spinning and the traditional ring-spinning. Photoshop was used to process the cross-sectional images of the air-jet vortex spinning yarn, determining the outer layer and the inner layer of the yarns. The relationship between the ratio of inner and outer fibers of air-jet spun yarns and twists were analyzed. The research results show that due to the special structure of the air-jet vortex spinning yarn, the traditional twisting and untwisting method is not suitable for the air-jet vortex spinning yarn twist test. It is feasible to test the twist by the image of air jet vortex spinning yarn. There is a negative correlation between the ratio of the number of fibers in the inner and outer layers of the air-jet vortex spinning yarn and its twist.

Modeling method of cyber physical production system for ring spinning
YIN Shiyong, BAO Jinsong, TANG Shixi, YANG Yun
Journal of Textile Research. 2021, 42(02):  65-73.  doi:10.13475/j.fzxb.20201007809
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In light of the problems such as long process flow of ring spinning, large variations in fiber morphology, mixed production continuity and discreteness, and the diverse and complex relationships between data flow and control flow in the production process, the ring spinning cyber physical production system architecture was comprehensively studied. According to the "computation, communication, and control" characteristics of cyber physical systems, the ring spinning cyber physical production system was defined. On this basis, the fiber flow, data flow and control flow of ring spinning cyber physical production were analyzed, leading to the establishment of a ring spinning cyber-physical production system model encompassing "fiber flow-data flow-control flow". Using the model-based system engineering method, the SysML modeling language was used to specify requirements for the partial models and full model for the ring spinning cyber physical production system considering "fiber flow-data flow-control flow". The results show that the proposed modeling method can effectively establish a ring spinning cyber physical production system model, and has certain reference significance for the intelligent production of ring spinning.

Preparation and performance of polyester/silk woven heart valve
HUANG Di, LI Fang, LI Gang
Journal of Textile Research. 2021, 42(02):  74-79.  doi:10.13475/j.fzxb.20201008207
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In order to develop an artificial heart valve with required mechanical properties and blood-proof performance, polyester (PET) multifilament and degummed silk fibroin (SF) were used for preparing woven artificial heart valve fabric (AHVF) through optimized design experiments on a sample weaving machine, with different structures, yarn linear densities and fabric densities. The optimization results show that when the thickness of AHVF is as thin as (0.52±0.1) mm, the AHVF demonstrates good hydro-philicity with contact angel of (60°±1.2°) which is closed to human heart valve. It is shown that AHVF has anisotropic mechanical properties and low permeability, with warp elastic modulus being 60~100 MPa, warp breaking strength 20~40 MPa, weft elastic modulus 7~50 MPa, weft breaking strength 7.5~20 MPa, and water permeability lower than 300 mL/(cm 2·min). Such properties of AHVF meet the standards for heart valve materials, indicating the feasibility of the AHVF for heart valve application.

Characterization of fabric smoothness appearance based on buckling-induced mechanical test with multiple deformation of fabrics
XIAO Caiqin, SUN Fengxin, GAO Weidong
Journal of Textile Research. 2021, 42(02):  80-86.  doi:10.13475/j.fzxb.20200601307
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The existing methods for testing fabric smoothness appearance based on image methods or visual principles are susceptible to the complex patterns and textures of fabrics, with shortcomings such as poor adaptability for different types of fabrics. Characterization method of fabric smoothness appearance was proposed based on the simultaneous-mechanical testing technology. 30 fabric samples were selected for in-situ mechanical test under multiple deformations, and five characteristic indexes of the force-displacement test curve were extracted respectively. SPSS software was used to correlate the extracted characteristic indexes, and fabric thickness with the subjective evaluation results. The stepwise regression method was used to construct a prediction model for fabric smoothness appearance, and the prediction model was validated with the help of a set of independent experiments. The results show that the evaluation results of the prediction model based on the in-situ mechanical test method have a good agreement with the subjective evaluation results, indicating the feasibility and effectiveness of the in-situ mechanical test method to characterize the appearance of the fabric.

Simulation and virtual display for few-guide bar yarn dyed fabric based Web
LIU Haisang, JIANG Gaoming, DONG Zhijia
Journal of Textile Research. 2021, 42(02):  87-92.  doi:10.13475/j.fzxb.20201000807
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In order to simulate and virtually display the few-guide bar yarn dyed fabrics, and analyze the knitting principle and the structure characteristics, based on the measurement and analysis of the actual stitch size, geometric models for stitches were established. Matrices for coordinates of stitches and transformation were adopted to obtain the stitch coordinates of the whole fabric according to the chain notation. Double buffer technology in the Web software was used to store the canvases with stitches and underlaps, which were superimposed together to implement the simulation. The pattern texture was captured as a screenshot, which was displayed on garment models by texture mapping. The simulated results were validated by comparing bitmaps of the real fabrics and the simulated ones. This research indicates that the method adopted is able to precisely simulate the pattern and size to achieve the visual design and display.

Threshold and intensity evaluation of skin wetness perception under dynamic contact with fabrics
ZHANG Zhaohua, TANG Xiangning, LI Jun, LI Luyao
Journal of Textile Research. 2021, 42(02):  93-100.  doi:10.13475/j.fzxb.20201005008
Abstract ( 173 )   HTML ( 5 )   PDF (3357KB) ( 119 )
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To gain insight into how fabrics affect the perception of wetness under dynamic skin contact at different velocities, the influencing mechanism of absolute threshold and intensity of the perception of wetness were investigated. By applying quantitative amounts of water (low, medium, and high) to each of the testing fabrics, participants reported the intensity of perceived wetness on a psychometric scale. In addition, water was supplied continually with a pump until the threshold of wetness was perceived by the participants. At the same time, the temperature sensors were used to record local skin temperature and calculate skin cooling rate. The results indicate that skin cooling rate has a significant positive correlation with wetness intensity rating, while a negative correlation with absolute threshold. The intensity rating of wetness perception is predicted by the physical parameters of the fabrics, that is maximum transient thermal flow, water content, and friction coefficient, while wetness threshold was predicted by wetting time and coefficient of friction. The threshold detection was qualified to evaluate the sensitivity to wetness at the initial detection of moisture on the skin, while the stimulus intensity rating would give a better prediction at the moisture absorption stage. This study provides the evaluation technology for designing clothing with desirable wetness levels.

Detection of fabric density uniformity based on convolutional neural network
MENG Shuo, XIA Xuwen, PAN Ruru, ZHOU Jian, WANG Lei, GAO Weidong
Journal of Textile Research. 2021, 42(02):  101-106.  doi:10.13475/j.fzxb.20201008406
Abstract ( 214 )   HTML ( 9 )   PDF (4877KB) ( 90 )
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Aiming at the narrow field of vision, low precision, and low adaptability in the current automatic methods for measuring fabric density, a multi-scale convolutional neural network was proposed. An offline image acquisition system was designed to acquire fabric images continuously, and a fabric image dataset containing detailed fabric parameters was established. Then, a multi-scale convolutional neural network with different sizes of local receptive fields was used for dealing with different fabric structure features and for locating yarns. The Hough transform and gray projection method were used to process the predicted yarn position map in order to calculate the warp and weft density and evaluate the fabric density uniformity. The results show that, compared with other methods, the detecting error of the warp and weft densities for different types of fabrics is less than 2%, which indicates that the proposed method has a higher accuracy and stronger variety adaptability.

Simulation on tensile properties of tubular weft knitted fabrics based on ABAQUS
SUN Yabo, LI Lijun, MA Chongqi, WU Zhaonan, QIN Yu
Journal of Textile Research. 2021, 42(02):  107-112.  doi:10.13475/j.fzxb.20200706106
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In order to better understand the tensile deformation capability of tubular weft knitted fabrics, a three-dimensional weft knitted fabric loop model and a tubular weft knitted fabric model were established using a three-dimensional modeling software Rhino, on the basis of the measured fabric structure parameters. The tensile properties of the fabric were simulated on unit loop level and tubular fabric level using the finite element analysis software ABAQUS. Numerical simulations of fabric tensile process were studied and corresponding experiment verification was carried out. Deformation and stress distribution of yarns in the stretching process of knitted fabrics were analyzed and discussed. The results indicate that when fabrics are stretched along the length direction, the finite element analysis results of its deformation and stress change are described accurately compared to the experimental results. Moreover, the difference in stress and strain values between the simulated and the experimental results is about 8%, indicating that the numerical simulation is feasible and effective.

Dyeing and Finishing & Chemicals
Simulation study on effect of organic solvents on dispersion and hydrolytic stability of liquid reactive dyes
XU Baolü, WU Wei, ZHONG Yi, XU Hong, MAO Zhiping
Journal of Textile Research. 2021, 42(02):  113-121.  doi:10.13475/j.fzxb.20200701109
Abstract ( 167 )   HTML ( 7 )   PDF (10850KB) ( 39 )
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In order to explore the mechanism of organic solvents affecting the storage stability of reactive dyes in liquid reactive dyes system, the effects of four organic solvents on the aggregation and hydrolysis of high concentration C.I. Reactive Blue 176 were investigated and verified by experiments, based on molecular dynamics simulation. Simulation analysis on radial distribution function of organic solvent molecules around dye shows that the difference in the effect of different organic solvents on the dispersion and hydrolysis of reactive dyes is related to the distribution of organic solvents in the effective area of dye molecules aggregation and around the hydrolytic groups of dyes. The strength of the interaction between them determines the storage stability of dyes. In N-methyl pyrrolidone systems, among 72 dyestuffs of the four organic solvents, 31 has the best dispersing effect on C.I. Reactive Blue 176, and the degree of dye aggregation is low, while caprolactam has the most effective effect on inhibiting the hydrolysis of liquid reactive dyes.

Fluorescence detection of Cr(VI) from printing and dyeing wastewater by zirconium-organic framework
GUAN Binbin, LI Qing, CHEN Linghui, XU Yuting, FAN Zenglu
Journal of Textile Research. 2021, 42(02):  122-128.  doi:10.13475/j.fzxb.20201002307
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Considering the pollution caused by heavy metal chromium ions in printing and dyeing wastewater, zirconium chloride and 2,2'-biquinoline-4,4'-dicarboxylic acid (H2L) were used to synthesize Zr-metal organic framework (Zr-MOF). The structural stability in water and in strong acid/base environments and photoluminescence property of Zr-MOF were confirmed by powder X-ray diffraction and fluorescence tests. The results of fluorescence sensing detection of Cr(VI) ions show that Zr-MOF performs highly selective fluorescence quenching recognition and quantitative detection towards trace $CrO 4 2 -$ and $Cr_{2}O_{7}^{2-}$ ions in water, under the interference of a variety of mixed anion and ions. The detection limits are confirmed to be 6.704 and 11.232 mg/L for $CrO 4 2 -$ and $Cr_{2}O_{7}^{2-}$ ions, respectively. The photoluminescence and fluorescence quenching detection mechanisms of Zr-MOF are proposed. Furthermore, after 7 consecutive fluorescence detection cycles, the fluorescence emission intensity of Zr-MOF still retains at above 95%.

Preparation of highly hollow biomass-based activated carbon fiber and its adsorption property to methylene blue
CHENG Lüzhu, WANG Zongqian, WANG Dengfeng, SHEN Jiakun, LI Changlong
Journal of Textile Research. 2021, 42(02):  129-134.  doi:10.13475/j.fzxb.20200805106
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In order to improve the adsorption efficiency of activated carbon fiber to colored wastewater, the activated carbon fibers with highly hollow structures were prepared, using the calotropis gigantea fiber as the precursor, by phosphoric acid activation and high temperature carbonization. The morphology, chemistry and pore structures of the calotropis gigantean activated carbon fiber were characterized systematically by scanning electron microscopy-energy dispersive spectrometer, Fourier transform infrared spectrocopy methods. Meanwhile, the adsorption properties and mechanism of carbon fibers to methylene blue were analyzed. The results show that the average hollow degree of calotropis gigantea activated carbon fiber is more than 92%, and the fiber also exhibits a rough surface and develops mesoporous structures with the specific surface area and average pore diameter of 1 244.812 m 2/g and 3.744 nm, respectively. Moreover, the surface of carbon fiber is rich in elements O and P, which constitute the active surface of carbon fiber. The saturated adsorption capacity of the carbon fiber to methylene blue aqueous solution with initial mass concentration of 100 mg/L can reach 198.840 mg/g. The adsorption process to methylene blue follows the pseudo-second-order kinetic equation. The adsorption to methylene blue conforms to the Freundlich model, which is mainly multilayer adsorption.

Bonding fastness of magnetron sputtering nano-films with various textile substrates
LIU Mingxue, ZHAO Qian, WANG Xiaohui, LIU Qiongxi, SHAO Jianzhong
Journal of Textile Research. 2021, 42(02):  135-141.  doi:10.13475/j.fzxb.20201008308
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To address the problem that the magnetron sputtering nano-films on natural fiber substrates are prone to whole layer shedding, the rubbing fastness and soaping of the magnetron sputtering structural color films on different fiber substrates were systematically tested, and the effects of the initial moisture content and the low-temperature plasma surface modification on the bonding fastness of the magnetron sputtering films to the fiber substrates were investigated in detail. The underlying reasons for the different bonding fastness of the magnetron sputtering films to different fiber substrates were studied by means of scanning electron microscope analysis technique combined with the analysis on physical and chemical properties of different fiber materials. The results show that due to the softening point of the thermoplastic polyester fiber, the kinetic energy of the energetic magnetron sputtered particles is converted into thermal energy when they are deposited on the fiber surface, causing the polyester fibers to locally reach the softening point and bonding the sputtered particles. On the contrary, cotton and silk fibers have no softening point, so the fibers have no melt-bonding effect on the magnetron sputtering particles. In addition, as cotton and silk fibers have hygroscopic swelling properties, the intrusion of water molecules during the soaping process weakens the interaction force between the fibers and nanoparticles, causing the easy shedding of the magnetron sputtering nano-films.

Preparation and decolorization of rayon based zeoliticimidazolate framework functional material
LOU Yaya, WANG Jing, DONG Yanchao, WANG Chunmei
Journal of Textile Research. 2021, 42(02):  142-147.  doi:10.13475/j.fzxb.20200700106
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In order to improve the decolorization performance of zeoliticimidazolate framework material (ZIF-8) for dyes, reduced graphene oxide(rGO)/ZIF-8 rayon based functional materials were prepared by loading graphene oxide (GO) and ZIF-8 on rayon surface followed by reducing. The functional materials were characterized by scanning electron microscopy, X-ray diffraction, and Fourier transform infrared spectrometer. The photocatalytic degradation of Reactive Black KN-B dye by rGO/ZIF-8 rayon based functional material under different light sources was studied, and the photocatalytic mechanism was discussed. The results show that the decolorization ratio of rGO/ZIF-8 rayon based functional material for Reactive Black KN-B dye is 97.1%, and the superoxide anion radical (·O 2-) produced by the reaction between the functional material and sunlight are captured by dissolved oxygen in water, which is the main active substance to degrade Reactive Black KN-B. After being reused for three times, the decolorization ratio of Reactive Black KN-B dye remains to be more than 90%.

Adsorption performance of chitosan based hybrid aerogel on reactive dyes
HE Xuemei, MAO Haiyan, CAI Lu
Journal of Textile Research. 2021, 42(02):  148-155.  doi:10.13475/j.fzxb.20200800408
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In order to effectively remove reactive dyes in textile dyeing wastewater,and hence to reduce the impact on water environment and human health hazards, achitosan based hybrid aerogel (CS/WSK/PDA/TiO2) was prepared by solution blending and freeze-drying techniques, using chitosan(CS) polymer as the skeleton material, dopamine(WSK) as bio-adhesive agent, and chitin whisker(PDA) as cross-linking agent. The hybrid aerogel was used to absorb the Reactive Violet X-2R, Reactive Red KE-3B and Reactive Yellow M-3BE. The morphology and structure of the prepared chitosan hybrid aerogels were characterized by attenuated total reflection Fourier transform in frared spectroscopy and scanning electron microscope. The effects of different conditions on the adsorption performance of the hybrid aerogel were studied. The experimental results show that chitosan-based hybrid aerogels have lamellar, porous functional structure,and the nano titanium oxide is embedded in the chitosan based hybrid aerogel. When the adsorption conditions are set to 30 ℃ and neutral pH value,the adsorption capacity of CS/WSK/PDA/TiO2 aerogels for Reactive Violet X-2R, Reactive Red KE-3B and Reactive Yellow M-3RE Reaches 400 mg/g, 398.4 mg/g, and 404.9 mg/g, respectively. The adsorption kinetic results are well described by the pseudo-second order kinetic.

Color prediction model of compound filament
WANG Yujuan, WANG Jun
Journal of Textile Research. 2021, 42(02):  156-160.  doi:10.13475/j.fzxb.20201007905
Abstract ( 145 )   HTML ( 5 )   PDF (1939KB) ( 29 )
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In order to facilitate the computer-aided pre-spinning design of original color-matching yarns, a color prediction model was proposed based on the transmittance, reflectance and arrangement of the monofilament from the perspective of light transmission between monofilaments. Red, green, yellow, and blue colored films instead of monofilament were used. The films of any two colors were cut into different sizes, and 12 multifilament samples were made by superimposing the colored films. The mixed color of the sample was tested with datacolor colorimeter, and the color difference between the mixed color tested by the colorimeter and predicted by the model was calculated. The factors causing the color difference were discussed through Pearson correlation analysis. The results show that the average predicted color difference is 0.43 CMC (2∶1) color difference unit. The saturation and hue angle of the multifilament has a significant impact on the predicted color difference, showing that the predicted color difference decreased with the increase of the saturation of the multifilament, and increases with the increase of the hue angle of the multifilament.

Preparation of self-crosslinking fluorinated polyacrylate emulsion and its application properties
CAI Lu, KANG Jialiang, LÜ Cun, HE Xuemei
Journal of Textile Research. 2021, 42(02):  161-167.  doi:10.13475/j.fzxb.20200908807
Abstract ( 187 )   HTML ( 4 )   PDF (6524KB) ( 43 )
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In view of the risks of long-chain perfluoroalkyl compounds to the environment and human body, four types of self-crosslinking fluorinated polyacrylates with different crosslinking agent contents were prepared by emulsion polymerization in this research. The long-chain perfluoroalkyl compounds were replaced by using environmentally friendly perfluorohexyl ethyl acrylate as functional monomer and γ-methacryloxypropyltrimethoxysilane as the crosslinking agent. The structure and performance of the self-crosslinking fluorinated polyacrylate emulsions were analyzed using particle size tester, Zeta potential analyzer, Fourier transform infrared spectrum and thermo-gravimetric analysis, and the application performance of self-crosslinking fluorinated polyacrylate emulsions were investigated with on cotton fabrics as the application object. The results show that the addition of crosslinking agent works in improving the thermal stability of fluorinated polyacrylate.Self-crosslinking fluorinated polyacrylate forms a complete water-repellent film on the surface of cotton fabric, and its static contact angle reaches as high as 138.2°. As the content of the crosslinking agent is further increased, the static contact angle of the treated cotton fabric drops to 90.6°.When the amount of crosslinking agent is 1.68 g, the treated cotton fabric demonstrats good washability.

Construction of superhydrophobic surface of cotton fabrics via dissolving etching
HAO Shang, XIE Yuan, WENG Jiali, ZHANG Wei, YAO Jiming
Journal of Textile Research. 2021, 42(02):  168-173.  doi:10.13475/j.fzxb.20201008807
Abstract ( 143 )   HTML ( 3 )   PDF (5169KB) ( 23 )
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In order to explor a green, simple and effective preparation method for making superhydrophobic cotton fabrics, polydimethylsiloxane (PDMS) coating was used to finish a cotton fabric with low surface energy, and then the salt particles were deposited and dissolved to form a rough structure. The effects of the amount of PDMS, salt particle size, deposition time and lasting time of the superhydrophobic properties of the fabric were investigated. The microstructure, element composition and stability of the superhydrophobic surface were characterized by of contact angle measuring instrument, scanning electron microscope and X-ray diffraction energy spectrum. The results show that there are micrometer pits on the surface of the cotton fabric. The static water contact angle (WCA) on the fabric surface is found up to 155.47° with the water slip angle of 5.5°. The WCA of the fabric, after immersing in strong acid and strong alkali solution for 12 h, still reaches 143.91°. After soaking the treated fabric in 60 ℃ water for 60 min, the WCA is 144.43°, and the WAC of the fabric drops by 11.31% only after 20 rubbing cycles. In addition, the superhydrophobic surface displays the self-cleaning function and anti-dyeing effect.

Preparation of multifunctional core-shell structure thermoelectric fabrics by low-temperature interfacial polymerization
ZHANG Xuefei, LI Tingting, SHIU Bingchiuan, LIN Jiahorng, LOU Chingwen
Journal of Textile Research. 2021, 42(02):  174-179.  doi:10.13475/j.fzxb.20200801206
Abstract ( 164 )   HTML ( 8 )   PDF (6397KB) ( 29 )
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In order to prepare high-conductivity, flexible, and multifunctional thermoelectric fabrics, a low-temperature in-situ interfacial polymerization method was proposed to fabricate core-shell thermoelectric textile with p-toluenesulfonic acid ion-doped poly(3,4-ethylenedioxythiophene) (PEDOT∶Tos) coated with polypropylene(PP) fibers. The structure and performance of thermoelectric fabrics were characterized and analyzed by scanning electron microscope, Fourier transform infrared spectrometer, and infrared thermal imager. The results show that the prepared thermoelectric fabrics have excellent flexibility as a textile materials, and good conductivity due to PEDOT∶Tos with conductivity reaching 2.1 S/cm. When a voltage of 10V is applied to both ends of a thermoelectric fabric, the surface temperature increases by about 20 ℃, indicating good electric heating performance and effective conversion of electric energy into heat energy. When the thermoelectric conversion device constructed using the thermoelectric fabric is placed in a temperature gradient field with a temperature difference of 20 ℃, it can continuously output a voltage of 0.3 mV.

Machinery & Accessories
Numerical analysis on formation mechanism of airflow field in rotor spinning unit
SHI Qianqian, WANG Jiang, ZHANG Yuze, LIN Huiting, WANG Jun
Journal of Textile Research. 2021, 42(02):  180-184.  doi:10.13475/j.fzxb.20201008705
Abstract ( 132 )   HTML ( 7 )   PDF (8140KB) ( 71 )
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The airflow field in a rotor spinning unit under a normal working condition is mainly affected by the air suction mechanism and rotor rotation mechanism. In order to investigate the contribution of the two mechanisms to the formation of airflow field in the rotor spinning unit, three cases corresponding to different operating conditions were established for investigation, and the fluid domain in the three cases based on the computational fluid dynamics were numerically simulated. The velocity distribution and air pressure distribution of the airflow field in three cases were also analyzed and discussed. Numerical simulation results show that the airflow field in the rotor spinning unit is determined by the air suction at rotor outlet and the high-speed rotor rotation. The air suction mechanism provides the necessary air velocity and negative pressure environment for fiber's transportation. The rotation mechanism assists in smooth transfer of the fibers to the rotor slide wall, the ordered arrangement of fibers, and the accumulation of the fibers to rotor groove. It is under the joint action of the two mechanisms that a unique spinning environment where fibers are driven using air for rotor spinning is formed.

Optimization design of electrode plate based on electrostatic adsorption and transfer used for garment fabric
LIU Lidong, LI Xinrong, LIU Hanbang, LI Dandan
Journal of Textile Research. 2021, 42(02):  185-192.  doi:10.13475/j.fzxb.20200802408
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In order to improve the effect of robot in the process of garment customization, unstable adsorption of the robot end-effector during grabbing and transferring fabric was worked on. The key factors of electrostatic plate design, the shape arrangement and structure parameters of the electrode plate were analyzed and optimized. The model of knitted weft knitted fabric was constructed, the four electrode distribution forms were simulated and compared, and the most suitable shape arrangement for garment fabric of the electrode plate: the comb electrode was obtained. The shape arrangement and structural parameters of the electrode plate were then optimized with the objective of maximizing the adsorption capacity per unit area. The simulation results show that the electrode with comb-shaped arrangement and optimized structure parameters can grab garment fabrics well, meeting the requirements for automatic garment production.

Transverse vibration characteristics of knitting needles in seamless underwear machines based on axial movement cantilever beam theory
DAI Ning, PENG Laihu, HU Xudong, ZHONG Yaosen, QI Dongming
Journal of Textile Research. 2021, 42(02):  193-20.  doi:10.13475/j.fzxb.20200803708
Abstract ( 150 )   HTML ( 6 )   PDF (4891KB) ( 34 )
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During the axial movement of the knitting needle in seamless underwear machines, the transverse vibration characteristics of the knitting needle are closely related to the performance of the knitting mechanism. However, there are few studies on the transverse vibration of knitting needles at this stage during the knitting process, hindering the improvement of stability of the seamless underwear machine in terms of vibration. In this paper, the vibration characteristics were theoretically studied with the knitting needle as an equivalence to an axially moving cantilever beam and the finite element calculation was carried out using ANSYS simulation software. The transverse vibration characteristics of the knitting needles were examined using a high-speed camera. The transverse vibration curve and dynamic frequency value calculated theoretically and by simulation demonstrates high level of resemblance with each other for the transverse vibration curve, and the maximum deviation of the dynamic frequency value is below 3.84%, The experimental results prove the existence of transverse vibration during the knitting process, and the frequency value obtained is consistent with the theoretical and simulation results, which implies the correctness of the theoretical and simulation modeling.


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# Section 6: Identity

Consider this sentence:

35. Pavel owes money to everyone else.

Let the UD be people; this will allow us to translate ‘everyone’ as a universal quantiﬁer. Let $$Oxy$$ mean ‘$$x$$ owes money to $$y$$’, and let $$p$$ mean Pavel. Now we can symbolize sentence 35 as ∀$$xOpx$$. Unfortunately, this translation has some odd consequences. It says that Pavel owes money to every member of the UD, including Pavel; it entails that Pavel owes money to himself. However, sentence 35 does not say that Pavel owes money to himself; he owes money to everyone else. This is a problem, because ∀$$xOpx$$ is the best translation we can give of this sentence into QL.

The solution is to add another symbol to QL. The symbol ‘=’ is a two-place predicate. Since it has a special logical meaning, we write it a bit diﬀerently: For two terms $$t$$1 and $$t$$2, $$t$$1 = $$t$$2 is an atomic formula.

The predicate $$x$$ = $$y$$ means ‘$$x$$ is identical to $$y$$.’ This does not mean merely that $$x$$ and $$y$$ are indistinguishable or that all of the same predicates are true of them. Rather, it means that $$x$$ and $$y$$ are the very same thing.

When we write $$x$$ ≠ $$y$$, we mean that $$x$$ and $$y$$ are not identical. There is no reason to introduce this as an additional predicate. Instead, $$x$$ ≠ $$y$$ is an abbreviation of ¬($$x$$ = $$y$$).

Now suppose we want to symbolize this sentence:

36. Pavel is Mister Checkov.

Let the constant $$c$$ mean Mister Checkov. Sentence 36 can be symbolized as $$p$$ = $$c$$. This means that the constants $$p$$ and $$c$$ both refer to the same guy.

This is all well and good, but how does it help with sentence 35? That sentence can be paraphrased as, ‘Everyone who is not Pavel is owed money by Pavel.’ This is a sentence structure we already know how to symbolize: ‘For all $$x$$, if $$x$$ is not Pavel, then $$x$$ is owed money by Pavel.’ In QL with identity, this becomes ∀$$x$$($$x$$ ≠ $$p$$ → $$Opx$$).

In addition to sentences that use the word ‘else’, identity will be helpful when symbolizing some sentences that contain the words ‘besides’ and ‘only.’ Consider these examples:

37. No one besides Pavel owes money to Hikaru.
38. Only Pavel owes Hikaru money.

We add the constant $$h$$, which means Hikaru.

Sentence 37 can be paraphrased as, ‘No one who is not Pavel owes money to Hikaru.’ This can be translated as ¬∃$$x$$($$x$$ ≠ $$p$$ & $$Oxh$$).

Sentence 38 can be paraphrased as, ‘Pavel owes Hikaru and no one besides Pavel owes Hikaru money.’ We have already translated one of the conjuncts, and the other is straightforward. Sentence 38 becomes $$Oph$$&¬∃$$x$$($$x$$≠ $$p$$ & $$Oxh$$).

## Expressions of quantity

We can also use identity to say how many things there are of a particular kind. For example, consider these sentences:

39. There is at least one apple on the table.
40. There are at least two apples on the table.
41. There are at least three apples on the table.

Let the UD be things on the table, and let Ax mean ‘$$x$$ is an apple.’

Sentence 39 does not require identity. It can be translated adequately as ∃$$xAx$$: There is some apple on the table— perhaps many, but at least one.

It might be tempting to also translate sentence 40 without identity. Yet consider the sentence ∃$$x$$∃$$y$$($$Ax$$&$$Ay$$). It means that there is some apple $$x$$ in the UD and some apple $$y$$ in the UD. Since nothing precludes $$x$$ and $$y$$ from picking out the same member of the UD, this would be true even if there were only one apple. In order to make sure that there are two diﬀerent apples, we need an identity predicate. Sentence 40 needs to say that the two apples that exist are not identical, so it can be translated as ∃$$x$$∃$$y$$($$Ax$$&$$Ay$$ & $$x$$ ≠ $$y$$).

Sentence 41 requires talking about three diﬀerent apples. It can be translated as ∃$$x$$∃$$y$$∃$$z$$($$Ax$$ & $$Ay$$ & $$Az$$ & $$x$$ ≠ $$y$$ & $$y$$ ≠ $$z$$ & $$x$$ ≠ $$z$$).

Continuing in this way, we could translate ‘There are at least $$n$$ apples on the table.’ There is a summary of how to symbolize sentences like these on p. 151.

Now consider these sentences:

42. There is at most one apple on the table.
43. There are at most two apples on the table.

Sentence 42 can be paraphrased as, ‘It is not the case that there are at least two apples on the table.’ This is just the negation of sentence 40:

¬∃$$x$$∃$$y$$($$Ax$$&$$Ay$$ & $$x$$ ≠ $$y$$)

Sentence 42 can also be approached in another way. It means that any apples that there are on the table must be the selfsame apple, so it can be translated as ∀$$x$$∀$$y$$[($$Ax$$&$$Ay$$) → $$x$$ = $$y$$]. The two translations are logically equivalent, so both are correct.

In a similar way, sentence 43 can be translated in two equivalent ways. It can be paraphrased as, ‘It is not the case that there are three or more distinct apples’, so it can be translated as the negation of sentence 41. Using universal quantiﬁers, it can also be translated as

∀$$x$$∀$$y$$∀$$z$$[($$Ax$$&$$Ay$$ &$$Az$$) → ($$x$$ = $$y$$∨$$x$$ = $$z$$∨$$y$$ = $$z$$)].

The examples above are sentences about apples, but the logical structure of the sentences translates mathematical inequalities like $$a$$ ≥ 3, $$a$$ ≤ 2, and so on. We also want to be able to translate statements of equality which say exactly how many things there are. For example:

44. There is exactly one apple on the table.
45. There are exactly two apples on the table.

Sentence 44 can be paraphrased as, ‘There is at least one apple on the table, and there is at most one apple on the table.’ This is just the conjunction of sentence 39 and sentence 42: ∃$$xAx$$&∀$$x$$∀$$y$$[($$Ax$$&$$Ay$$) → $$x$$ = $$y$$]. This is a somewhat complicated way of going about it. It is perhaps more straightforward to paraphrase sentence 44 as, ‘There is a thing which is the only apple on the table.’ Thought of in this way, the sentence can be translated ∃$$x$$[$$Ax$$&¬∃$$y$$($$Ay$$ & $$x$$ ≠$$y$$)].

Similarly, sentence 45 may be paraphrased as, ‘There are two diﬀerent apples on the table, and these are the only apples on the table.’ This can be translated as ∃$$x$$∃$$y$$[$$Ax$$&$$Ay$$ & $$x$$ ≠ $$y$$ &¬∃$$z$$($$Az$$ & $$x$$ ≠ $$y$$ & $$y$$ ≠ $$z$$)].

Finally, consider this sentence:

46. There are at most two things on the table.

It might be tempting to add a predicate so that $$Tx$$ would mean ‘$$x$$ is a thing on the table.’ However, this is unnecessary. Since the UD is the set of things on the table, all members of the UD are on the table. If we want to talk about a thing on the table, we need only use a quantiﬁer. Sentence 46 can be symbolized like sentence 43 (which said that there were at most two apples), but leaving out the predicate entirely. That is, sentence 46 can be translated as ∀$$x$$∀$$y$$∀$$z$$($$x$$ = $$y$$∨$$x$$ = $$z$$∨$$y$$ = $$z$$).

Techniques for symbolizing expressions of quantity (‘at most’, ‘at least’, and ‘exactly’) are summarized on p. 151.

## Deﬁnite descriptions

Recall that a constant of QL must refer to some member of the UD. This constraint allows us to avoid the problem of non-referring terms. Given a UD that included only actually existing creatures but a constant $$c$$ that meant ‘chimera’ (a mythical creature), sentences containing $$c$$ would become impossible to evaluate.

The most widely inﬂuential solution to this problem was introduced by Bertrand Russell in 1905. Russell asked how we should understand this sentence:

47. The present king of France is bald.

The phrase ‘the present king of France’ is supposed to pick out an individual by means of a deﬁnite description. However, there was no king of France in 1905 and there is none now. Since the description is a non-referring term, we cannot just deﬁne a constant to mean ‘the present king of France’ and translate the sentence as $$Kf$$.

Russell’s idea was that sentences that contain deﬁnite descriptions have a different logical structure than sentences that contain proper names, even though they share the same grammatical form. What do we mean when we use an unproblematic, referring description, like ‘the highest peak in Washington state’? We mean that there is such a peak, because we could not talk about it otherwise. We also mean that it is the only such peak. If there was another peak in Washington state of exactly the same height as Mount Rainier, then Mount Rainier would not be the highest peak.

According to this analysis, sentence 47 is saying three things. First, it makes an existence claim: There is some present king of France. Second, it makes a uniqueness claim: This guy is the only present king of France. Third, it makes a claim of predication: This guy is bald.

In order to symbolize deﬁnite descriptions in this way, we need the identity predicate. Without it, we could not translate the uniqueness claim which (according to Russell) is implicit in the deﬁnite description.

Let the UD be people actually living, let $$Fx$$ mean ‘$$x$$ is the present king of France’, and let $$Bx$$ mean ‘$$x$$ is bald.’ Sentence 47 can then be translated as ∃$$x$$[$$Fx$$&¬∃$$y$$($$Fy$$ & $$x$$ ≠ $$y$$)&$$Bx$$]. This says that there is some guy who is the present king of France, he is the only present king of France, and he is bald.

Understood in this way, sentence 47 is meaningful but false. It says that this guy exists, but he does not.

The problem of non-referring terms is most vexing when we try to translate negations. So consider this sentence:

48. The present king of France is not bald.

According to Russell, this sentence is ambiguous in English. It could mean either of two things:

48a. It is not the case that the present king of France is bald.
48b. The present king of France is non-bald.

Both possible meanings negate sentence 47, but they put the negation in different places.

Sentence 48a is called a wide-scope negation, because it negates the entire sentence. It can be translated as ¬∃$$xFx$$&¬∃$$y$$($$Fy$$ & $$x$$ ≠ $$y$$)&$$Bx$$. This does not say anything about the present king of France, but rather says that some sentence about the present king of France is false. Since sentence 47 if false, sentence 48a is true.

Sentence 48b says something about the present king of France. It says that he lacks the property of baldness. Like sentence 47, it makes an existence claim and a uniqueness claim; it just denies the claim of predication. This is called narrow-scope negation. It can be translated as ∃$$xFx$$&¬∃$$y$$($$Fy$$ & $$x$$ ≠ $$y$$)&¬$$Bx$$. Since there is no present king of France, this sentence is false.

Russell’s theory of deﬁnite descriptions resolves the problem of non-referring terms and also explains why it seemed so paradoxical. Before we distinguished between the wide-scope and narrow-scope negations, it seemed that sentences like 48 should be both true and false. By showing that such sentences are ambiguous, Russell showed that they are true understood one way but false understood another way.

For a more detailed discussion of Russell’s theory of deﬁnite descriptions, including objections to it, see Peter Ludlow’s entry ‘descriptions’ in The Stanford Encyclopedia of Philosophy: Summer 2005 edition, edited by Edward N. Zalta, http://plato.stanford.edu/archives/s.../descriptions/

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# Sharing a simple and surprising mathematical idea from “How to gamble if you must” with kids

Yesterday I got a neat book in the mail – I’d seen Nassim Taleb recommend it on Twitter:

One warning – this is not a “popular math” book, it is pretty math heavy. Flipping through the first 1/4 of the book, I really enjoyed the presentation and was once again reminded of the surprising fact that foundational research on basic gambling problems was being done in the late 1950s and early 1960s. An accessible and incredibly interesting account of some of this work can be found in Ed Thorp’s autobiography “A man for all Markets.”

One nice example from the beginning of the book relates to gambling in a 50/50 game called “red and black.” Think of the game as trying to guess the color of a card pulled from a randomly shuffled deck, or just betting on a coin flip. If you want to turn, say, \$100 into \$1,000 by betting on this game, what is your best strategy?

IF you are interested, a shorter account of this problem (with accompanying practice problems) can be found in this nice summary paper by Kyle Seigrist published by the Mathematical Association of America.

Summary of the ideas from “How to gamble if you must” by Kyle Seigrist and the Mathematical Association of America

Because this particular gambling problem is accessible to kids, for today’s project I wanted to introduce the idea of 50/50 gambles and ask them what they thought the optimal gambling strategy would be. The specific question is what is the best strategy to follow if you want to try to turn \$100 into \$1,000?

They had some absolutely terrific ideas. My 6th grade son practically suggested the betting strategy from the Kelly criterion!

Next we turned to the computer to study this game in Mathematica. We looked at some simple betting ideas first. So, if we want to turn \$100 into \$1,000 in this game, what happens if we bet \$100 on each bet? What happens if we bet \$50 on each bet?

After seeing the surprising results from the fist set of trials, we looked at the gambling strategies that the boys proposed. First we looked at a version of the strategy that my old son suggested -> basically bet the maximum amount every time (except when you don’t need to bet the max amount to reach \$1,000).

Are you more or less likely to turn \$100 into \$1,000 with this strategy?

Now we checked the betting strategy that my younger son suggested -> bet 1/2 your money each time (except when you don’t need to bet that much to reach \$1,000).

The boys had some pretty interesting ideas about what would happen here.

So, definitely a fun project and the result is pretty surprising (at least to me!) -> in 50/50 games your betting strategy doesn’t matter.

# Christopher Long’s fun generalization of an Expii problem

Twitter is really great place to see fun math. Before showing the fun generalization, though, just to avoid spoilers I want to show the original problem. Here’s the direct link:

https://www.expii.com/solve/69/5

and here’s the problem itself:

So, I’ll pause here to not ruin the problem for anyone who wants to work on it.

pause

pause

pause

pause

pause

pause

pause

pause

pause

pause

Ok . . . here’s the really cool set of tweets I saw from Christopher Long this morning:

which continued as follows:

To see how delightful Long’s general solution is, maybe walking through my chicken scratch solution which happened to be still sitting on my desk will be helpful:

Here’s a sketch of my approach.

In order to maximize your chance of winning a bet like the one in the problem (one that you expect to lose) you should bet as much as you can at each step (subject to a maximum of the amount you need to win) at each stage.

So,

(i) at step 1 the probability of getting the tree to grow to 40 feet is 1/5 and probability of losing is 4/5.

(ii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 80 feet and a 4/5 probability of losing.

(iii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 100 feet and a 4/5 probability of having it shrink to 60 feet.

(iv) If you win on stage (iii) you win (1 out of 125 times). If you lose, you now have a 1/5 probability of having the tree now grow to 100 feet and a 4/5 probability of having the tree shrink to 20 feet.

So, after the 4th branch in my picture you’ve either won (probability 1/125 + 4/625), returned to 20 feet (probability 16/625) or lost (the only other case).

Thus, your probability of winning the game from the start, $x$, satisfies the equation:

$x = 1/125 + 4/625 + (16/625)*x$

We can solve this pretty easily to see that $x = 3/203$.

The really fun – and honestly, amazing – thing about Long’s solution is that he notices that the pattern in the branches of the binary tree corresponds exactly to the pattern in the digits of the binary expansion of 1/5. For clarity, the 1/5 here comes from the growth multiple – 20 feet growing to 100 feet – and not from the probability which, by coincidence, also has a 1/5 in it.

Anyway, Long’s solution also allows you to immediately see how to solve any problem like the Expii one, and, for extra fun, problems where the growth multiple is irrational:

The answer is in Long’s timeline, but it is a good challenge to see if you can work out the answer just from the tweets I’ve included here. Since he skips a bit of algebra in his tweets, working through his tweets is also an important way to make sure that you really understand his work.

I think the sequence of tweets from Long are a great thing to show kids who are learning math – especially kids learning probability and stats. Those tweets really show how a mathematician thinks about a problem.

# A probability problem from James Tanton’s “Solve This”

We were unpacking a bunch of books today and ran across James Tanton’s Solve This and Theoni Pappaas’s The Adventures of Penrose The Mathematical Cat. I asked the boys to find a problem they’d like to talk about and they found a gem from Tanton’s book:

16.1 A Fair Game?

Peter has ten coins, Pennelope has nine. Peter and Pennelope agree to toss all their coins simultaneously. Whoever receives the largest number of heads will win. In case of a tie Pennelope will be declared the winner, so as to offset the advantage Peter has to begin with. Given this agreement, who is most likely to win?

We began by talking about the problem and the boys had some ideas right away. Their initial guess is that Peter will win, but they want to explore the problem by looking at an easier case first.

After discussing the problem we started looking at the easier case in which Peter has one coin and Pennelope has 1 coin. One of the nice things about this simple version of the problem is that you can actually talk through all of the cases. Talking through these cases is a really nice introduction to counting and probability. The other nice thing about the simple case, of course, is the surprising answer!

Given the surprising answer from the last video, we decided to check out the next easiest case – Peter has 3 coins and Pennelope has 2. Fortunately we’ve already understood the cases that Pennelope will encounter here, so we just have to understand the situations that can come up for Peter.

Again, this exercise is a nice exercise in counting and also this time the snap cubes we are using for heads and tails help show the symmetry involved in the various cases. The boys do a nice job here working through this slightly more complicated example:

Finally we decided to try to tackle the original problem. My younger son has a great symmetry idea what we talk about for a bit, but it seemed that approaching the problem this way was still a bit out of reach for them.

They wanted to calculate, but I didn’t 🙂 As sort of an on-the-fly compromise, I thought working a little bit with Pascal’s triangle would still be a good exercise and also show them that the calculation they were looking to do would be pretty ugly!

So, maybe not the most satisfying end to the project since we didn’t completely solve the problem. Nonetheless, I like all of the great math practice that we got in this project – lots of good ideas from introductory probability and counting, and lots of good arithmetic practice, too.

Also, it was nice to see the boys starting to think about using symmetry to help solve problems. Even if they aren’t yet quite able to put those ideas to work in their solution yet, I’m glad those ideas are starting to come to the surface.

Tomorrow we are starting to work through Art of Problem Solving’s Introduction to Counting and Probability. Today’s project was a great way to motivate that summer project!

# The triangle inequality and geometric probability

I finished up a section on the triangle inequality today with my older son and wanted to show him a surprising application. This example – taken from geometric probability – is a little over his head, but was still fun to talk through.

The problem is this:

You have a stick with length 1. If you chop the stick into 3 pieces, what is the probability that those three pieces can form a triangle?

Fun question – we started out with him giving his thoughts on the problem, and the problem of thinking through probability when you have an infinite number of choices:

With the initial thoughts out of the way, we started thinking through a few specific ways to cut the stick into pieces. The important observation from this part of the talk is that if any piece has length greater than 1/2, we won’t be able to form a triangle:

Finally we get to the geometric probability. We have to start with a special property of an equilateral triangle. We’d talked about this property of an equilateral triangle yesterday. My son remembered the property, but not the proof. We spent a minute or two reviewing the proof, but the proof wasn’t the main point.

We can use an equilateral triangle with height of 1 to model our probability problem in a surprising way. With our new model, we see that the amazing connection to geometry tells us that the probability of our broken up stick forming a triangle is 1/4.

So, a fun little project that is a little more advanced than what we’ve covered so far. It touches on geometric probability which is an incredible subject on its own, it touches a little on properties of infinity, and finally the relationship between algebra and geometry.

All from the simple ideas behind the triangle inequality. Amazing!


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# Real k-flats tangent to quadrics in R^n

Research paper by Frank Sottile, Thorsten Theobald

Indexed on: 31 Jan '05Published on: 31 Jan '05Published in: Mathematics - Algebraic Geometry

#### Abstract

Let d_{k,n} and #_{k,n} denote the dimension and the degree of the Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each k between 1 and n-2 there are 2^{d_{k,n}} \cdot #_{k,n} (a priori complex) k-planes in P^n tangent to d_{k,n} general quadratic hypersurfaces in P^n. We show that this class of enumerative problem is fully real, i.e., for each k between 1 and n-2 there exists a configuration of d_{k,n} real quadrics in (affine) real space R^n so that all the mutually tangent k-flats are real.


text:
# Cochran's Theorem - Lemma involving symmetric idempotent matrices

I am writing notes for a reading class and I've decided to add a proof of Cochran's theorem in order to show that a statistic is $\chi^2$. I am struggling for the proof of a particular lemma but the rest is just peachy.

Lemma: Let $A$ be a real real symmetric idempotent matrix of order $n$ with rank $r$. Suppose $A=A_1+\cdots+A_k$ with rank $A_i = r_i$ and $A_i$ symmetric. Additionally, $r_1+\cdots+r_k=r$. Then each $A_i$ is idempotent.

I've been really struggling with this. It seems fairly obvious that $A_i$ gives an orthogonal decomposition of $A$. I've tried a lot of methods, looking at $A$'s and $A_i'$s spectral decomposition so I can show that the eigenvalues of $A_i$ must be $1$. Tried showing that $A_iA_j =0$ for $i$ not equal to $j$.

-

The ranks of the matrices are the dimensions of their column spaces. Since the ranks of the $A_i$ add up to the rank of $A$, the column space $V$ of $A$ must be the direct sum of the column spaces $V_i$ of the $A_i$. Since $A$ is idempotent, it acts as the identity on $V$, and thus in particular on the $V_i$. That is, applying $A$ to a vector $v_i$ in $V_i$ leaves that vector invariant. Since each $A_i$ only contributes a component in its column space, it follows that $A_iv_i=v_i$ and $A_jv_i=0$ for $i\ne j$. Thus each $A_i$ is the identity on its column space and zero on its complement, and thus idempotent.
I am not certain. In the context of statistics the $A_i$'s are quadratic forms of random variables. I wonder if, at any rate, they are necessarily symmetric. My linear algebra is a bit rusty. – user18263 Oct 26 '11 at 7:05


text:
Toward a Characterization of Loss Functions for Distribution Learning

Nika Haghtalab, Cameron Musco, Bo Waggoner

In this work we study loss functions for learning and evaluating probability distributions over large discrete domains. Unlike classification or regression where a wide variety of loss functions are used, in the distribution learning and density estimation literature, very few losses outside the dominant $log\ loss$ are applied. We aim to understand this fact, taking an axiomatic approach to the design of loss functions for learning distributions. We start by proposing a set of desirable criteria that any good loss function should satisfy. Intuitively, these criteria require that the loss function faithfully evaluates a candidate distribution, both in expectation and when estimated on a few samples. Interestingly, we observe that \emph{no loss function} possesses all of these criteria. However, one can circumvent this issue by introducing a natural restriction on the set of candidate distributions. Specifically, we require that candidates are $calibrated$ with respect to the target distribution, i.e., they may contain less information than the target but otherwise do not significantly distort the truth. We show that, after restricting to this set of distributions, the log loss, along with a large variety of other losses satisfy the desired criteria. These results pave the way for future investigations of distribution learning that look beyond the log loss, choosing a loss function based on application or domain need.

arrow_drop_up


text:
# Synopsis: Ordering in hydrogen under high pressure

Molecular dynamics studies indicate a new phase of liquid hydrogen under high pressure.

The behavior of hydrogen under pressure affects fields ranging from condensed matter physics to astrophysics. Compressed liquid hydrogen exhibits a molecular-to-atomic transition. However, despite several experiments, the theoretical debate on the nature of this transition, in particular whether it is a continuous or discontinuous (first-order) transition, has not yet been settled.

In an article in Physical Review Letters, Isaac Tamblyn and Stanimir Bonev of Dalhousie University in Nova Scotia, Canada, employ molecular dynamics to map the phase diagram of dense hydrogen over a large range of temperatures and pressures. Their findings suggest an unreported phase in the liquid with short-ranged orientational order, resulting from a transition that they predict should occur in the liquid above $100\phantom{\rule{0.333em}{0ex}}\text{GPa}$. This new phase may explain certain characteristics of the molecular-to-atomic transition, the shape of the melting line, as well as the structure of hydrogen mixtures. The authors argue that a first-order transition is likely, and the new insight provided should spur future experimental work. – Sami Mitra

### Announcements

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text:
# Parameterized 3D Heat Sink¶

## Introduction¶

In this tutorial, we will walk through the process of simulating a parameterized problem using Modulus. The neural networks in Modulus allow us to solve problems for multiple parameters/independent variables in a single training. These parameters can be geometry variables, coefficients of a PDE or even boundary conditions. Once the training is complete, it is possible to run inference on several geometry/physical parameter combinations as a post-processing step, without solving the forward problem again. We will see that such parameterization increases the computational cost only fractionally while solving the entire desired design space.

To demonstrate this feature we will solve the flow and heat over a 3-fin heat sink whose fin height, fin thickness, and fin length are variable. We will then perform a design optimization to find out the most optimal fin configuration for our heat sink example. By the end of this tutorial, you would learn to easily convert any simulation to a parametric design study using Modulus’s CSG module and Neural Network solver. In this tutorial, you would learn the following:

1. How to set up a parametric simulation in Modulus.

Note

This tutorial is an extension of tutorial Conjugate Heat Transfer where we discussed how to use Modulus for solving Conjugate Heat problems. In this tutorial, we take up the same geometry setup and solve it for a parameterized setup at an increased Reynolds number. Hence, we recommend you to refer tutorial Conjugate Heat Transfer for any additional details related to geometry specification and boundary conditions.

The same scripts used in example Conjugate Heat Transfer will be used. To make the simulation parameterized and turbulent, we will set the custom flags parameterized and turbulent both as true in the config files.

Note

In this tutorial we will focus on parameterization which is independent of the physics being solved and can be applied to any class of problems covered in the User Guide.

## Problem Description¶

Please refer the geometry and boundary conditions for a 3-fin heat sink in tutorial Conjugate Heat Transfer. We will parameterize this problem to solve for several heat sink designs in a single neural network training. We will modify the heat sink’s fin dimensions (thickness, length and height) to create a design space of various heat sinks. The Re for this case is now 500 and we incorporate turbulence using Zero Equation turbulence model.

For this problem, we will vary the height ($$h$$), length ($$l$$), and thickness ($$t$$) of the central fin and the two side fins. The height, length, and thickness of the two side fins are kept the same, and therefore, there will be a total of six geometry parameters. The ranges of variation for these geometry parameters are given in equation (198).

(198)$\begin{split}\begin{split} h_{central fin} &= (0.0, 0.6),\\ h_{side fins} &= (0.0, 0.6),\\ l_{central fin} &= (0.5, 1.0) \\ l_{side fins} &= (0.5, 1.0) \\ t_{central fin} &= (0.05, 0.15) \\ t_{side fins} &= (0.05, 0.15) \end{split}\end{split}$

## Case Setup¶

In this tutorial, we will use the 3D geometry module from Modulus to create the parameterized 3-fin heat sink geometry. Discrete parameterization can sometimes lead to discontinuities in the solution making the training harder. Hence in this tutorial we will only cover parameters that are continuous. Also, we will be training the parameterized model and validating it by performing inference on a case where $$h_{central fin}=0.4$$, $$h_{side fins}=0.4$$, $$l_{central fin}=1.0$$, $$l_{side fins}=1.0$$, $$t_{central fin}=0.1$$, and $$t_{side fins}=0.1$$. At the end of the tutorial we will also present the comparison between results for the above combination of parameters obtained from a parameterized model versus results obtained from a non-parameterized model trained on just a single geometry corresponding to the same set of values. This will highlight the usefulness of using PINNs for doing parameterized simulations in comparison to some of the traditional methods.

Since the majority of the problem definition and setup was covered in Conjugate Heat Transfer, we will focus only on important elements for the parameterization.

### Creating Nodes and Architecture for Parameterized Problems¶

The parameters chosen for variables act as additional inputs to the neural network. The outputs remain the same. Also, for this example since the vairables are geometric only, no change needs to be made for how the equation nodes are defined (except the addition of turbulence model). In cases where the coefficients of a PDE are parameterized, the corresponding coefficient needs to be defined symbolically (i.e. using string) in the equation node.

Note for this example, the viscosity is set as a string in the NavierStokes constructor for the purposes of turbulence model. The ZeroEquation equation node 'nu' as the output node which acts as input to the momentum equations in Navier-Stokes.

Below we show the code for this parameterized problem. Note that parameterized and turbulent are set to true in the config file.

Parameterized flow network:

    # make navier stokes equations
if cfg.custom.turbulent:
ns = NavierStokes(nu="nu", rho=1.0, dim=3, time=False)
ze = ZeroEquation(nu=0.002, dim=3, time=False, max_distance=0.5)
navier_stokes_nodes = (
ns.make_nodes()
+ ze.make_nodes()
+ [Node.from_sympy(geo.sdf, "normal_distance")]
)
else:
ns = NavierStokes(nu=0.01, rho=1.0, dim=3, time=False)
navier_stokes_nodes = ns.make_nodes()
normal_dot_vel = NormalDotVec()

# make network arch
if cfg.custom.parameterized:
input_keys = [
Key("x"),
Key("y"),
Key("z"),
Key("fin_height_m"),
Key("fin_height_s"),
Key("fin_length_m"),
Key("fin_length_s"),
Key("fin_thickness_m"),
Key("fin_thickness_s"),
]
else:
input_keys = [Key("x"), Key("y"), Key("z")]
flow_net = FullyConnectedArch(
input_keys=input_keys, output_keys=[Key("u"), Key("v"), Key("w"), Key("p")]
)

# make list of nodes to unroll graph on
flow_nodes = (
navier_stokes_nodes
+ normal_dot_vel.make_nodes()
+ [flow_net.make_node(name="flow_network", jit=False)]
)

Parameterized heat network:

# make network arch
if cfg.custom.parameterized:
input_keys = [Key("x"), Key("y"), Key("z"), Key('fin_height_m'), Key('fin_height_s'), Key('fin_length_m'), Key('fin_length_s'), Key('fin_thickness_m'), Key('fin_thickness_s')]
else:
input_keys = [Key("x"), Key("y"), Key("z")]
flow_net = FullyConnectedArch(
input_keys=input_keys,
output_keys=[Key("u"), Key("v"), Key("w"), Key("p")],
)
thermal_f_net = FullyConnectedArch(
input_keys=input_keys,
output_keys=[Key("theta_f")]
)
thermal_s_net = FullyConnectedArch(
input_keys=input_keys,
output_keys=[Key("theta_s")]
)

# make list of nodes to unroll graph on
thermal_nodes = (
+ dif.make_nodes()
+ dif_inteface.make_nodes()
+ [flow_net.make_node(name="flow_network", optimize=False)]
+ [thermal_f_net.make_node(name="thermal_f_network")]
+ [thermal_s_net.make_node(name="thermal_s_network")]
)

### Setting up Domain and Constraints¶

This section is again very similar to Conjugate Heat Transfer tutorial. The only difference being, now the input to param_ranges argument is a dictionary of key value pairs where the keys are strings for each design variable and the values are tuples of float/ints specifying the range of variation for those variables.

The code to setup these dictionaries for parameterized inputs and constraints can be found below.

Setting the parameter ranges (three_fin_geometry.py)

# parametric variation
fin_height_m, fin_height_s = Symbol('fin_height_m'), Symbol('fin_height_s')
fin_length_m, fin_length_s = Symbol('fin_length_m'), Symbol('fin_length_s')
fin_thickness_m, fin_thickness_s = Symbol('fin_thickness_m'), Symbol('fin_thickness_s')
height_m_range = (0.0, 0.6)
height_s_range = (0.0, 0.6)
length_m_range = (0.5, 1.0)
length_s_range = (0.5, 1.0)
thickness_m_range = (0.05, 0.15)
thickness_s_range = (0.05, 0.15)
param_ranges = {fin_height_m: height_m_range,
fin_height_s: height_s_range,
fin_length_m: length_m_range,
fin_length_s: length_s_range,
fin_thickness_m: thickness_m_range,
fin_thickness_s: thickness_s_range}
fixed_param_ranges = {fin_height_m: 0.4,
fin_height_s: 0.4,
fin_length_m: 1.0,
fin_length_s: 1.0,
fin_thickness_m: 0.1,
fin_thickness_s: 0.1}

Setting the param_ranges argument in the constraints. Here we only show the example for a few BCs from the flow domain. But the same settings are applied to all the other BCs.

    # set param ranges
if cfg.custom.parameterized:
pr = param_ranges
else:
pr = fixed_param_ranges

# params for simulation
# fluid params
inlet_vel = 1.0
volumetric_flow = 1.0

# make flow domain
flow_domain = Domain()

# inlet
constraint_inlet = PointwiseBoundaryConstraint(
nodes=flow_nodes,
geometry=inlet,
outvar={"u": inlet_vel, "v": 0, "w": 0},
batch_size=cfg.batch_size.Inlet,
criteria=Eq(x, channel_origin[0]),
lambda_weighting={
"u": channel.sdf,
"v": 1.0,
"w": 1.0,
},  # weight zero on edges
param_ranges=pr,
batch_per_epoch=5000,
)

    # integral continuity
integral_continuity = IntegralBoundaryConstraint(
nodes=flow_nodes,
geometry=integral_plane,
outvar={"normal_dot_vel": volumetric_flow},
batch_size=5,
integral_batch_size=cfg.batch_size.IntegralContinuity,
criteria=geo.sdf > 0,
lambda_weighting={"normal_dot_vel": 1.0},
param_ranges={**pr, **{x_pos: (-1.1, 0.1)}},
fixed_dataset=False,
num_workers=4,
)

## Training the Model¶

This part is exactly similar to tutorial Conjugate Heat Transfer and once all the definitions are complete, we can execute the parameterized problem like any other problem.

## Design Optimization¶

As discussed previously, you can optimize the design once the training is complete as a post-processing step. A typical design optimization usually contains an objective function that is minimized/maximized subject to some physical/design constraints.

For heat sink designs, usually the peak temperature that can be reached at the source chip is limited. This limit arises from the operating temperature requirements of the chip on which the heat sink is mounted for cooling purposes. The design is then constrained by the maximum pressure drop that can be successfully provided by the cooling system that pushes the flow around the heat sink. Mathematically this can be expressed as below:

 Variable/Function Description minimize $$Peak \text{ } Temperature$$ Minimize the peak temperature at the source chip with respect to $$h_{central fin}, h_{side fins}, l_{central fin}, l_{side fins}, t_{central fin}, t_{side fins}$$ Geometric Design variables of the heat sink subject to $$Pressure \text{ } drop < 2.5$$ Limit on the pressure drop (Max pressure drop that can be provided by cooling system

Such optimization problems can be easily achieved in Modulus once you have a trainied, parameterized model ready. As it can be noticed, while solving the parameterized simulation we created some monitors to track the peak temperature and the pressure drop for some design variable combination. We basically would follow the same process and use the PointwiseMonitor constructor to find the values for multiple combinations of the design variables. You can create this simply by looping through the multiple designs. Since these monitors can be for large number of design variable combinations, we recommend to use these monitors only after the training is complete to achieve better computational efficiency. Do do this, once the models are trained, you can run the flow and thermal models in the 'eval' mode by specifying run_mode: 'run_mode=eval' in the config files.

After the models are run in the 'eval' mode, the pressure drop and peak temperature values will be saved in form of a .csv file. Then, one can write a simple scripts to sift through the various samples and pick the most optimal ones that minimize/maximize the objective function while meeting the required constraints (for this example, the design with the least peak temperature and the maximum pressure drop < 2.5):

"""
NOTE: run three_fin_flow and Three_fin_thermal in "eval" mode
after training to get the monitor values for different designs.
"""

# import Modulus library
from modulus.csv_utils.csv_rw import dict_to_csv
from modulus.hydra import to_absolute_path

# import other libraries
import numpy as np
import os, sys
import csv

# specify the design optimization requirements
max_pressure_drop = 2.5
num_design = 10
path_flow = to_absolute_path("outputs/three_fin_flow")
path_thermal = to_absolute_path("outputs/three_fin_thermal")
invar_mapping = [
"fin_height_middle",
"fin_height_sides",
"fin_length_middle",
"fin_length_sides",
"fin_thickness_middle",
"fin_thickness_sides",
]
outvar_mapping = ["pressure_drop", "peak_temp"]

# read the monitor files, and perform a design space search
def DesignOpt(
path_flow,
path_thermal,
num_design,
max_pressure_drop,
invar_mapping,
outvar_mapping,
):
path_flow += "/monitors"
path_thermal += "/monitors"
directory = os.path.join(os.getcwd(), path_flow)
sys.path.append(path_flow)
values, configs = [], []

for _, _, files in os.walk(directory):
for file in files:
if file.startswith("back_pressure") & file.endswith(".csv"):
value = []
configs.append(file[13:-4])

with open(os.path.join(path_flow, file), "r") as datafile:
data = []
columns = [row[1]]
data.append(columns)
last_row = float(data[-1][0])
value.append(last_row)

with open(
os.path.join(path_flow, "front_pressure" + file[13:]), "r"
) as datafile:
data = []
columns = [row[1]]
data.append(columns)
last_row = float(data[-1][0])
value.append(last_row)

with open(
os.path.join(path_thermal, "peak_temp" + file[13:]), "r"
) as datafile:
data = []
columns = [row[1]]
data.append(columns)
last_row = float(data[-1][0])
value.append(last_row)
values.append(value)

# perform the design optimization
values = np.array(
[
[values[i][1] - values[i][0], values[i][2] * 273.15]
for i in range(len(values))
]
)
indices = np.where(values[:, 0] < max_pressure_drop)[0]
values = values[indices]
configs = [configs[i] for i in indices]
opt_design_index = values[:, 1].argsort()[0:num_design]
opt_design_values = values[opt_design_index]
opt_design_configs = [configs[i] for i in opt_design_index]

# Save to a csv file
opt_design_configs = np.array(
[
np.array(opt_design_configs[i][1:].split("_")).astype(float)
for i in range(num_design)
]
)
opt_design_configs_dict = {
key: value.reshape(-1, 1)
for (key, value) in zip(invar_mapping, opt_design_configs.T)
}
opt_design_values_dict = {
key: value.reshape(-1, 1)
for (key, value) in zip(outvar_mapping, opt_design_values.T)
}
opt_design = {**opt_design_configs_dict, **opt_design_values_dict}
dict_to_csv(opt_design, "optimal_design")
print("Finished design optimization!")

if __name__ == "__main__":
DesignOpt(
path_flow,
path_thermal,
num_design,
max_pressure_drop,
invar_mapping,
outvar_mapping,
)

## Results¶

The design parameters for the optimal heat sink for this problem are: $$h_{central fin} = 0.4$$, $$h_{side fins} = 0.4$$, $$l_{central fin} = 0.83$$, $$l_{side fins} = 1.0$$, $$t_{central fin} = 0.15$$, $$t_{side fins} = 0.15$$. The above design has a pressure drop of 2.46 and a peak temperature of 76.23 $$(^{\circ} C)$$ Fig. 123

Table 7 represents the computed pressure drop and peak temperature for the OpenFOAM single geometry and Modulus single and parameterized geometry runs. It is evident that the results for the parameterized model are close to those of a single geometry model, showing its good accuracy.

 Property OpenFOAM Single Run Single Run Parameterized Run Pressure Drop $$(Pa)$$ 2.195 2.063 2.016 Peak Temperature $$(^{\circ} C)$$ 72.68 76.10 77.41

By parameterizing the geometry, Modulus significantly accelerates design optimization when compared to traditional solvers, which are limited to single geometry simulations. For instance, 3 values (two end values of the range and a middle value) per design variable would result in $$3^6 = 729$$ single geometry runs. The total compute time required by OpenFOAM for this design optimization would be 4099 hrs. (on 20 processors). Modulus can achieve the same design optimization at ~17x lower computational cost. Large number of design variables or their values would only magnify the difference in the time taken for two approaches.

Note

The Modulus calculations were done using 4 Nvidia V100 GPUs. The OpenFOAM calculations were done using 20 processors.

Here, the 3-Fin heatsink was solved for arbitrary heat properties chosen such that the coupled conjugate heat transfer solution was possible. However, such approach causes issues when the conductivities are orders of magnitude different at the interface. We will revisit the conjugate heat transfer problem in tutorial Heat Transfer with High Thermal Conductivity and Industrial Heat Sink to see some advanced tricks/schemes that one can use to handle the issues that arise in Neural network training when real material properties are involved.