text:
CONTACT INFO

Dr. Chenfeng Li 1, 2

 1 Zienkiewicz Centre for Computational Engineering Post: College of Engineering, Swansea University Bay Campus, Swansea SA1 8EN, UK. 2 Energy Safety Research Institute Tel: +44-01792-602256 College of Engineering Email: c.f.li@swansea.ac.uk

SPECIALIST SUBJECTS

• Computational Engineering
• Computer Graphics
• Civil and Structural Engineering
• Geo-mechanics, Oil & Gas Reservoir
• Concrete, Composite, Heterogeneous Materials
• RESEARCH INTERESTS

• Computer simulation of deformation, fracture, energy and mass transport
• Physically-based modelling with visual applications
• Uncertainty quantification, reliability and risk assessment
•

JOURNAL Papers

2016

Multiphase SPH simulation for interactive fluids and solids
ACM Transactions on Graphics (SIGGRAPH 2016), 35 (4) (2016), Article No. 79.
X. Yan, Y.T. Jiang, C.F. Li, R.R. Martin and S.M. Hu

This work extends existing multiphase-fluid SPH frameworks to cover solid phases, including deformable bodies and granular materials. In our extended multiphase SPH framework, the distribution and shapes of all phases, both fluids and solids, are uniformly represented by their volume fraction functions. The dynamics of the multiphase system is governed by conservation of mass and momentum within different phases. The behavior of individual phases and the interactions between them are represented by corresponding constitutive laws, which are functions of the volume fraction fields and the velocity fields. Our generalized multiphase SPH framework does not require separate equations for specific phases or tedious interface tracking. As the distribution, shape and motion of each phase is represented and resolved in the same way, the proposed approach is robust, efficient and easy to implement. Various simulation results are presented to demonstrate the capabilities of our new multiphase SPH framework, including deformable bodies, granular materials, interaction between multiple fluids and deformable solids, flow in porous media, and dissolution of deformable solids.

Statistical reconstruction and Karhunen-Loève expansion for multiphase random media
International Journal for Numerical Methods in Engineering, 105 (1) (2016) 3-32.
J.W. Feng, S. Cen, C.F. Li and D.R.J. Owen

Termed as random media, rocks, composites, alloys and many other heterogeneous materials consist of multiple material phases that are randomly distributed through the medium. This paper presents a robust and efficient algorithm for reconstructing random media, which can then be fed into stochastic finite element solvers for statistical response analysis. The new method is based on nonlinear transformation of Gaussian random fields, and the reconstructed media can meet the discrete-valued marginal probability distribution function and the two-point correlation function of the reference medium. The new method, which avoids iterative root-finding computation, is highly efficient and particularly suitable for reconstructing large-size random media or a large number of samples. Also, benefiting from the high efficiency of the proposed reconstruction scheme, a Karhunen–Loève (KL) representation of the target random medium can be efficiently estimated by projecting the reconstructed samples onto the KL basis. The resulting uncorrelated KL coefficients can be further expressed as functions of independent Gaussian random variables to obtain an approximate Gaussian representation, which is often required in stochastic finite element analysis.

Automatic sizing functions for unstructured surface mesh generation
International Journal for Numerical Methods in Engineering, 109 (4) (2016) 577-608.
J.J. Chen, Z.F. Xiao, Y. Zheng, J.J. Zheng, C.F. Li and K.W. Liang

Accurate sizing functions are crucial for efficient generation of high-quality meshes, but to define the sizing function is often the bottleneck in complicated mesh generation tasks because of the tedious user interaction involved. We present a novel algorithm to automatically create high-quality sizing functions for surface mesh generation. First, the tessellation of a Computer Aided Design (CAD) model is taken as the background mesh, in which an initial sizing function is defined by considering geometrical factors and user-specified parameters. Then, a convex nonlinear programming problem is formulated and solved efficiently to obtain a smoothed sizing function that corresponds to a mesh satisfying necessary gradient constraint conditions and containing a significantly reduced element number. Finally, this sizing function is applied in an advancing front mesher. With the aid of a walk-through algorithm, an efficient sizing-value query scheme is developed. Meshing experiments of some very complicated geometry models are presented to demonstrate that the proposed sizing-function approach enables accurate and fully automatic surface mesh generation.

Finite element analysis for inclined well bore stability of transversely isotropic rock with HMCD coupling based on weak plane strength criterion
Science China , 59 (2016) 1-14.
Y.L. Wang, Z. Zhuang, Z.L. Liu, H.L. Yang and C.F. Li

The finite element analysis (FEA) technology by hydraulic-mechanical-chemical-damage (HMCD) coupling is proposed in this paper for inclined well bore stability analysis of water-sensitive and laminated rock, developed basing on the recently established FEA technology for transversely isotropic rock with hydraulic-mechanical-damage (HMD) coupling. The chemical activity of the drilling fluid is considered as phenomenological hydration behavior, the moisture content and parameters of rock considering hydration could be determined with time. The finite element (FE) solutions of numerical well bore model considering the chemical activity of drilling fluid, damage tensor calculation and weak plane strength criterion for transversely isotropic rock are developed for researching the well bore failure characteristics and computing the time-dependent collapse and fracture pressure of laminated rock as shale reservoirs. A three-dimensional FE model and elastic solid deformation and seepage flow coupled equations are developed, and the damage tensor calculation technology for transversely isotropic rock are realized by introducing effect of the hydration and the stress state under the current load. The proposed method utilizing weak plane strength criterion fully reflects the strength parameters in rock matrix and weak plane. To the end, an effective and reliable numerically three-step FEA strategy is well established for well bore stability analysis. Numerical examples are given to show that the proposed method can establish efficient and applicable FE model and be suitable for analyzing the time-dependent solutions of pore pressure and stresses, and the evolution region considering the hydration surrounding well bore, furthermore to compute the collapse cycling time and the safe mud weight for collapse and fracture pressure of transversely isotropic rock.

An unsymmetrical 8-node hexahedral element with high distortion tolerance
International Journal for Numerical Methods in Engineering, 109 (8) (2016) 1130-1158.
P.L. Zhou, J.B. Huang, C.F. Li, Q. Zhang and S. Cen

Among all 3D 8-node hexahedral solid elements in current finite element library, the ‘best’ one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8-node, 24-DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetrical element stiffness matrix. The first set simply utilizes the formulations of the traditional 8-node trilinear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US-ATFH8, contains no adjustable factor and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first-order (constant stress/strain) patch test and the second-order patch test for pure bending, remove the volume locking, and provide the invariance for coordinate rotation. Especially, it is insensitive to various severe mesh distortions.

A 4-node quadrilateral flat shell element formulated by the shape-free HDF plate and HSF membrane elements
Engineering Computations, 33 (3) (2016) 713-741.
Y. Shang, S. Cen and C.F. Li

Purpose – The purpose of this paper is to propose an efficient low-order quadrilateral flat shell element that possesses all outstanding advantages of novel shape-free plate bending and plane membrane elements proposed recently. Design/methodology/approach – By assembling a shape-free quadrilateral hybrid displacement-function (HDF) plate bending element HDF-P4-11ß (Cen et al., 2014) and a shape-free quadrilateral hybrid stress-function (HSF) plane membrane element HSF-Q4?-7ß (Cen et al., 2011b) with drilling degrees of freedom (DOF), a new 4-node, 24-DOF (six DOFs per node) quadrilateral flat shell element is successfully constructed. The trial functions for resultant/stress fields within the element are derived from the analytical solutions of displacement and stress functions for plate bending and plane problems, respectively, so that they can a priori satisfy the related governing equations. Furthermore, in order to take the influences of moderately warping geometry into consideration, the rigid link correction strategy (Taylor, 1987) is also employed. Findings – The element stiffness matrix of a new simple 4-node, 24-DOF quadrilateral flat shell element is obtained. The resulting models, denoted as HDF-SH4, not only possesses all advantages of original HDF plate and HSF plane elements when analyzing plate and plane structures, but also exhibits good performances for analyses of complicated spatial shell structures. Especially, it is quite insensitive to mesh distortions. Originality/value – This work presents a new scheme, which can take the advantages of both analytical and discrete methods, to develop low-order mesh-distortion resistant flat shell elements.

Fast SPH simulation for gaseous fluids
The Visual Computer, 32 (4) (2016) 523-534.
B. Ren, X. Yan, T. Yang, C.F. Li, M.C. Lin and S.M. Hu

This paper presents a fast smoothed particle hydro-dynamics (SPH) simulation approach for gaseous fluids. Unlike previous SPH gas simulators, which solve the transparent air flow in a fixed simulation domain, the proposed approach directly solves the visible gas without involving the transparent air. By compensating the density and force calculation for the visible gas particles, we completely avoid the need of computational cost on ambient air particles in previous approaches. This allows the computational resources to be exclusively focused on the visible gas, leading to significant performance improvement of SPH gas simulation. The proposed approach is at least ten times faster than the standard SPH gas simulation strategy and is able to reduce the total particle number by 25–400 times in large open scenes. The proposed approach also enables fast SPH simulation of complex scenes involving liquid–gas transition, such as boiling and evaporation. A particle splitting and merging scheme is proposed to handle the degraded resolution in liquid–gas phase transition. Various examples are provided to demonstrate the effectiveness and efficiency of the proposed approach.

Quasi-static crack propagation modeling using shape-free hybrid stress-function elements with drilling degrees of freedom
International Journal of Computational Methods, 13 (3) (2016) 1650014.
S. Cen, Y. Bao and C.F. Li

A new plane shape-free multi-node singular hybrid stress-function (HS-F) element with drilling degrees of freedom, which can accurately capture the stress intensity factors at the crack tips, is developed. Then, a quasi-static 2D crack propagation modeling strategy is established by combination of the new singular element and a shape-free 4-node HS-F plane element with drilling degrees of freedom proposed recently. Only simple re-meshing with an unstructured mesh is needed for each simulation step. Numerical results show that the proposed scheme is an effective and robust technique for dealing with the crack propagation problems.

2015

A simple approach for bubble modeling from multiphase fluid simulation
Computational Visual Media, 1 (2) (2015) 171-181.
B. Ren, Y.T. Jiang, C.F. Li, T. Yang and Ming C. Lin

This article presents a novel and flexible bubble modeling technique for multi-fluid simulations using a volume fraction representation. By combining the volume fraction data obtained from a primary multi-fluid simulation with simple and efficient secondary bubble simulation, a range of real-world bubble phenomena are captured with a high degree of physical realism, including large bubble deformation, sub-cell bubble motion, bubble stacking over the liquid surface, bubble volume change, dissolving of bubbles, etc. Without any change in the primary multi-fluid simulator, our bubble modeling approach is applicable to any multi-fluid simulator based on the volume fraction representation.

An unsymmetrical 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem
International Journal for Numerical Methods in Engineering, 103 (7) (2015) 469-500.
S. Cen, P.L. Zhou, C.F. Li and C.J. Wu

Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal’s theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a C0 patch test when the element’s shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetrical formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates .x; y/ and the second form of quadrilateral area coordinates (QACM-II) (S,T) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal’s theorem.

Automatic surface repairing, defeaturing and meshing algorithms based on an extended B-Rep
Advances in Engineering Software, 86 (2015) 55-69.
J.J. Chen, B.W. Cao, Y. Zheng, L.J. Xie, C.F. Li and Z.F. Xiao

This paper presents an extended surface boundary representation (B-rep), where each topology entity can have dual geometric representations to accommodate various defects (e.g., gaps and overlaps) commonly present in CAD models. Keeping a uniform B-rep and the unsuppressed geometry data enables the use of various existing repairing, defeaturing and meshing algorithms to process CAD models with small gaps and overlaps on surface boundaries. The continuous geometry of the input model remains untouched in the repairing, defeaturing and meshing process, and the output mesh is loyal to this geometry. Such feature is often desirable in numerical simulations that require meshes with high geometry
fidelity.

Two generalized conforming quadrilateral Mindlin-Reissner plate elements based on the displacement function
Finite Element in Analysis and Design, 99 (2015) 24-38.
Y. Shang, S. Cen, C.F. Li and X.R. Fu

This work presents two 4-node, 12-DOF quadrilateral displacement-based finite elements for analysis of the Mindlin-Reissner plate. Derived from the fundamental analytical solutions of the displacement function F, the deflection and rotation fields of the proposed elements satisfy a priori all related governing equations. The unknown coefficients are determined through the generalized conforming element method, a relaxed and rational conforming approach. The resulting elements perform like nonconforming elements on a coarse mesh, and with mesh refinement they converge as conforming elements. Numerical benchmarks demonstrate that the new elements are insensitive to mesh distortion and free of shear locking, and can provide satisfactory results for most cases.

An effective hybrid displacement function element method for solving the edge effect of Mindlin-Reissner plate
International Journal for Numerical Methods in Engineering, 102 (8) (2015) 1449-1487.
Y. Shang, S. Cen, C.F. Li and J.B. Huang

In bending problems of Mindlin-Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply-supported (SS1) boundaries. This is so-called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the hybrid displacement function (HDF) element method [1], a simple hybrid-Trefftz stress element method proposed recently. What is different is that an additional displacement function f related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF-P4-Free and HDF-P4-SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF-P4-11 \beta [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes.

A priori error estimation for the stochastic perturbation method
Computer Methods in Applied Mechanics and Engineering, 286 (2015) 1-21.
X.Y. Wang, S. Cen, C.F. Li and D.R.J. Owen

The perturbation method has been among the most popular stochastic finite element methods due to its simplicity and efficiency. The error estimation for the perturbation method is well established for deterministic problems, but until now there has not been an error estimation developed in the probabilistic context. This paper presents a priori error estimation for the perturbation method in solving stochastic partial differential equations. The physical problems investigated here come from linear elasticity of heterogeneous materials, where the material parameters are represented by stochastic fields. After applying the finite element discretization to the physical problem, a stochastic linear algebraic equation system is formed with a random matrix on the left hand side. Such systems have been efficiently solved by using the stochastic perturbation approach, without knowing how accurate/inaccurate the perturbation solution is. In this paper, we propose a priori error estimation to directly link the error of the solution vector with the variation of the source stochastic field. A group of examples are presented to demonstrate the effectiveness of the proposed error estimation.

2014

The mechanical behavior and reliability prediction of the HTR graphite component at various temperature and neutron dose ranges
Nuclear Engineering and Design 276 (2014) 9-18.
X. Fang, S.Y. Yu, H.T. Wang and C.F. Li

In a pebble-bed high temperature gas-cooled reactor (HTR), nuclear graphite serves as the main structuralmaterial of the side reflectors. The reactor core is made up of a large number of graphite bricks. Inthe normal operation case of the reactor, the maximum temperature of the helium coolant commonlyreaches about 750◦C. After around 30 years' full power operation, the peak value of in-core fast neutron-cumulative dose reaches to 1 × 1022n cm−2(EDN). Such high temperature and neutron irradiation strongly impact the behavior of graphite component, causing obvious deformation. The temperature and neutron-dose are unevenly distributed inside a graphite brick, resulting in stress concentrations. The deformation and stress concentration can both greatly affect safety and reliability of the graphite component. In addition, most of the graphite properties (such as Young's modulus and coefficient of thermal expansion)change remarkably under high temperature and neutron irradiations. The irradiation-induced creep also plays a very important role during the whole process, and provides a significant impact on the stress accumulation. In order to simulate the behavior of graphite component under various in-core conditions,all of the above factors must be considered carefully. In this paper, the deformation, stress distribution and failure probability of a side graphite component are studied at various temperature points and neutron-dose levels. 400◦C, 500◦C, 600◦C and 750◦C are selected as the core-side temperature of the graphite component, while the range of neutron dose is 0–1 × 1022n cm−2(EDN). It is shown in the numerical-analysis that temperature strongly affects the histories of both stresses and failure probabilities as fast-neutron dose increases. In addition, the behavior of graphite brick at 750◦C is observed obviously different from that at other temperatures.

A quasi-static crack propagation simulation based on shape-free hybrid stress-function finite elements with simple re-meshing
Computer Methods in Applied Mechanics and Engineering 275 (2014) 159-188.
M.J. Zhou, S. Cen, Y. Bao and C.F. Li

In this paper, a new shape-free multi-node singular hybrid stress-function (HSF) element and a shape-free 8-node plane HSF element proposed recently are employed to simulate the quasi-static 2D crack propagation problem. Compared with other well-known methods, such new scheme exhibits four advantages: (i) for the singular element, the shape and the number of nodes can be flexibly adjusted as required; (ii) high precision for stress intensity factors (SIF) can be obtained due to the advantages of the HSF method; (iii) only simple remeshing with a very coarse mesh is needed for each simulation step; (iv) unstructured mesh containing extremely distorted elements can be used without losing precision. It demonstrates that the proposed scheme is an effective technique for dealing with crack propagation problems.

Multiple-fluid SPH Simulation Using a Mixture Model
ACM Transactions on Graphics / SIGGRAPH, 33 (5) (2014) 171.
R. Bo, C.F. Li, X. Yan, M.C. Lin, J. Bonet and S.M. Hu

This paper presents a versatile and robust SPH simulation approach for multiple-fluid flows. The spatial distribution of different phases or components is modeled using the volume fraction representation, the dynamics of multiple-fluid flows is captured by using an improved mixture model, and a stable and accurate SPH formulation is rigorously derived to resolve the complex transport and transformation processes encountered in multiple-fluid flows. The new approach can capture a wide range of realworld multiple-fluid phenomena, including mixing/unmixing of miscible and immiscible fluids, diffusion effect and chemical reaction etc. Moreover, the new multiple-fluid SPH scheme can be readily integrated into existing state-of-the-art SPH simulators, and the multiple-fluid simulation is easy to set up. Various examples are presented to demonstrate the effectiveness of our approach.

Statistical reconstruction of two-phase random media
Computers and Structures, 137 (2014) 78-92.
J.W. Feng, C.F. Li, S. Cen and D.R.J. Owen

A robust and efficient algorithm is proposed to reconstruct two-phase composite materials with random morphology, according to given samples or given statistical characteristics. The new method is based on nonlinear transformation of Gaussian random fields, where the correlation of the underlying Gaussian field is determined explicitly rather than through iterative methods. The reconstructed media can meet the binary-valued marginal probability distribution function and the two point correlation function of the reference media. The new method, whose main computation is completed using fast Fourier transform (FFT), is highly efficient and particularly suitable for reconstructing large size random media or a large number of samples. Its feasibility and performance are examined through a series of practical examples with comparisons to other state-of-the-art methods in random media reconstruction.

Hybrid displacement function element method, a modified hybrid-stress / hybrid-Trefftz element method for analysis of Mindlin-Reissner plate
International Journal for Numerical Methods in Engineering, 98 (2014) 203-234.
S. Cen, Y. Shang, C.F. Li and H.G. Li

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin-Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F . Third, the displacements and shear strains along each element boundary are determined by the locking-free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid-Trefftz stress element method. As an example, a 4-node, 12-DOF quadrilateral plate bending element, HDF-P4-11?, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape-free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed.

Artistic preprocessing for image painterly rendering and stylization
The Visual Computer, 30 (2014) 969-979.
Y. Zang, H. Huang and C.F. Li

A practical image enhancing technique is presented as a preprocessing step for painterly rendering and image stylization, which transforms the input image mimicking the vision of artists. The new method contains mainly two parts dealing with artistic enhancement and color adjustment, respectively. First, through feature extraction and simplification, an abstract shadow map is constructed for the input image, which is then taken as a guide for emphasizing the light–shadow contrast and the important shadow lines. Next, to simulate the intense color emotion often subjectively added by the artist, a color adjustment technique is proposed to generate lively colors with sharp contrast imitating the artistic vision. The preprocessing operation is compatible with existing stylization and stroke-based painterly rendering techniques, and it can also produce different types of stylization results independently.

2013

Generalized Neumann expansion and its application in stochastic finite element methods
Mathematical Problems in Engineering, Volume 2013 (2013), Article ID 325025, http://dx.doi.org/10.1155/2013/325025
X.Y. Wang, S. Cen and C.F. Li

An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNEmethod, the perturbationmethod, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumannmethod are obtained for the first time in the probability context.

Stroke style analysis for painterly rendering
Journal of Computer Science and Technology, 28 (5) (2013) 762-775.
Y. Zang, H. Huang and C.F. Li

We propose a novel method that automatically analyzes stroke-related artistic styles of paintings. A set of adaptive interfaces are also developed to connect the style analysis with existing painterly rendering systems, so that the specific artistic style of a template painting can be effectively transferred to the input photo with minimal effort. Different from conventional texture-synthesis based rendering techniques that focus mainly on texture features, this work extracts, analyzes and simulates high-level style features expressed by artists' brush stroke techniques. Through experiments, user studies and comparisons with ground truth, we demonstrate that the proposed style-orientated painting framework can significantly reduce tedious parameter adjustment, and it allows amateur users to efficiently create desired artistic styles simply by specifying a template painting.

Flow field modulation
IEEE Transaction on Visualization and Computer Graphics, 19 (10) (2013) 1708-1719.
B. Ren, C.F. Li, Ming C. Lin, T. Kim and S.M. Hu

The nonlinear and non-stationary nature of Navier-Stokes equations produces fluid flows that can be noticeably different in appearance with subtle changes. In this paper we introduce a method that can analyze the intrinsic multiscale features of flow fields from a decomposition point of view, by using the Hilbert-Huang transform method on 3D fluid simulation. We show how this method can provide insights to flow styles and help modulate the fluid simulation with its internal physical information.We provide easy-to-implement algorithms that can be integrated with standard grid-based fluid simulation methods, and demonstrate how this approach can modulate the flow field and guide the simulation with different flow styles. The modulation is straightforward and relates directly to the flow's visual effect, with moderate computational overhead.

View-dependent multiscale fluid simulation
IEEE Transaction on Visualization and Computer Graphics, 19 (2) (2013) 178-188.
Y. Gao, C.F. Li, B. Ren and S.M. Hu

Fluid flows are highly nonlinear and non-stationary, with turbulence occurring and developing at different length and time scales. In real-life observations, the multiscale flow generates different visual impacts depending on the distance to the viewer. We propose a new fluid simulation framework that adaptively allocates computational resources according to the viewer's position. First, a 3D empirical mode decomposition scheme is developed to obtain the velocity spectrum of the turbulent flow. Then, depending on the distance to the viewer, the fluid domain is divided into a sequence of nested simulation partitions. Finally, the multiscale fluid motions revealed in the velocity spectrum are distributed non-uniformly to these view-dependent partitions, and the mixed velocity fields defined on different partitions are solved separately using different grid sizes and time steps. The fluid flow is solved at different spatial-temporal resolutions, such that higher-frequency motions closer to the viewer are solved at higher resolutions and vice versa. The new simulator better utilizes the computing power, producing visually plausible results with realistic fine-scale details in a more efficient way. It is particularly suitable for large scenes with the viewer inside the fluid domain. Also, as high-frequency fluid motions are distinguished from low-frequency motions in the simulation, the numerical dissipation is effectively reduced.

2012

Failure probability study of HTR graphite component using microstructure-based model
Nuclear Engineering and Design, 253 (2012) 192-199.
S.Y. Yu, X. Fang, H.T. Wang and C.F. Li

Nuclear graphite is widely used as in-core structural material in the high temperature gas-cooled reactors (HTRs). As mechanical properties of graphite change with neutron irradiation and temperature, the reliability evaluation of graphite internals of a HTR is commonly accomplished by the finite element analysis using various constitutive creep models and the following structural failure prediction with introduction of failure models that have essential impact on the final evaluation result. In this paper, a microstructurebased failure model proposed by Burchell in 1996 in conjunction with the finite element code INET-GRA3D of INET is used to study failure probability of a graphite side reflector design in the pebble-bed HTR during its entire service life from the microstructural view. The corresponding irradiation-induced creep stress analysis is carried out using both the UKAEA model and the Kennedy model. H-451 graphite is selected as material input since its irradiation material data from both macrostructure tests and microstructure observation is available. The results are compared with those using the macrostructure-based Weibull model commonly accepted in the engineering field and consistent trends are observed. Moreover, The Burchell model leads to the failure probability more sensitive to stress levels than the Weibull model.

An evaluation on fatigue crack growth in a fine-grained isotropic graphite, Nuclear Engineering and Design
Nuclear Engineering and Design, 250 (2012) 197-206.
H.T. Wang, L.B. Sun, C.F. Li, H.T. Wang and L. Shi

The aim of this paper is to investigate the mechanism of fatigue crack propagation in IG-11 graphite, and determine the crack growth rate in relation to the stress level. Experimental studies were performed at both micro and macro scales. For fatigue microcrack propagation, single-edge-notch specimens were chosen for testing and the fatigue crack growth was measured in situ with a scanning electron microscope. For fatigue macrocrack propagation, CT specimens were used and the fatigue crack growth was measured with a high-accuracy optic microscope. Combining the two groups of experimental results, the following conclusions are derived: (1) The heterogeneous microstructures of the graphite material have significant impact on the fatigue microcrack growth, while their influence on fatigue macrocrack growth is very limited. (2) The relationship between the fatigue crack growth rate and the crack-tip stress intensity factor range can be expressed in the form of Paris formulae, which contains three stages: an initial rising part with a small slope, an abrupt rise with a very large acceleration, and a short final part with a small slope. (3) The fatigue fracture surface of the graphite material contains considerable sliding of leaf-shape graphite flakes combined with small cotton-shape plastic deformations. These sliding traces are approximately aligned at 45?, showing the main cause of the fatigue fracture is the shear stress. There are also a large amount of secondary cracks inside unit cells and on cell walls, which indicates the fracture mechanism of the IG-11 graphite is mainly cleavage fracture.

A novel joint diagonalization approach for linear stochastic systems and reliability analysis
Engineering Computations, 29 (2) (2012) 221-244.
F. Wang, C.F. Li, J.W. Feng, S. Cen and D.R.J. Owen

We present a novel gradient based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers. By starting with a least squares (LS) criterion, we obtain a classical nonlinear cost-function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications. A novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerical gradient base, and solves the problem with iterations. Demonstrated by examples, the new algorithm shows the merits of simplicity, effectiveness, and computational efficiency.

The various creep models for irradiation behavior of nuclear graphite
Nuclear Engineering and Design, 242 (2012) 19-25.
X. Fang, H.T. Wang, S.Y. Yu and C.F. Li

Graphite is a widely used material in nuclear reactors, especially in high temperature gascooled Q2 reactors (HTRs), in which it plays three main roles: moderator, reflector and structure material. Irradiationinduced creep has a significant impact on the behavior of nuclear graphite as graphite is used in high temperature and neutron irradiation environments. Thus the creep coefficient becomes a key factor in stress analysis and lifetime prediction of nuclear graphite. Numerous creep models have been established, including the visco-elastic model, UK model, and Kennedy model. A Fortran code based on user subroutines of MSC.MARC was developed in INET in order to perform three-dimensional finite element analysis of irradiation behavior of the graphite components for HTRs in 2008, and the creep model used is for the visco-elastic model only. Recently the code has been updated and can be applied to two other models—the UK model and the Kennedy model. In the present study, all three models were used for calculations in the temperature range of 280-450 ◦C and the results are contrasted. The associated constitutive law for the simulation of irradiated graphite covering properties, dimensional changes, and creep is also briefly reviewed in this paper. It is shown that the trends of stresses and life prediction of the three models are the same, but in most cases the Kennedy model gives the most conservative results while the UK model gives the least conservative results. Additionally, the influence of the creep strain ratio is limited, while the absence of primary creep strain leads to a great increase of failure probability.

Fisheye video correction
IEEE Transaction on Visualization and Computer Graphics, 18 (10) (2012) 1771-1783.
J. Wei, C.F. Li, S.M. Hu, R.R. Martin and C.L. Tai

Various types of video can be captured with fisheye lenses, particularly surveillance video, due to their ability to capture a wide field of view. However, fisheye lenses introduce distortion, which changes as objects in the scene move. This makes fisheye video difficult to interpret. Current still fisheye image correction methods are either limited to small angles of view, or are strongly content-dependent, and therefore unsuitable for processing video streams. We present an efficient and robust scheme for fisheye video correction, which minimizes time-varying distortions and preserves salient content in a coherent manner. Our optimization process is controlled by user annotation, and takes into account a wide set of measures addressing different aspects of natural scene appearance. Each is represented as a quadratic term in an energy minimization problem, leading to a closed form solution via a sparse linear system. We illustrate our method with a range of examples, demonstrating coherent natural-looking video output; the visual quality of individual frames is comparable to those produced by state-of-the-art methods for fisheye still photograph correction.

2011

Painterly rendering with content-dependent natural paint strokes
The Visual Computer, 27 (9) (2011) 861-871
H. Huang, T.N. Fu and C.F. Li

We present a new painterly rendering method that simulates artists' content-dependent painting process and the natural variation of hand-painted strokes. First, a new stroke layout strategy is proposed to enhance the contrast between large and small paint strokes, which is an important characteristic of hand-painted paintings. Specifically, the input image is partitioned into nonuniform grids according to its importance map, and determined by the grid size, an individually constructed paint stroke is applied in each grid. Second, an anisotropic digital brush is designed to simulate a real paint brush. In particular, each bristle of the digital brush has an individual color, so that strokes rendered by the new brush can have multiple colors and naturally varied textures. Finally, we present a novel method to add lighting effects to the canvas. This lighting imitation method is robust and very easy to implement, and it can significantly improve the quality of rendering. Comparing with traditional painterly rendering approaches, the new method simulates more closely the real painting procedure, and our experimental results show that it can produce vivid paintings with fewer artifacts.

Shape-free finite element method: The plane hybrid stress-function (HS-F) element method for anisotropic materials
Science China - Physics, Mechanics & Astronomy, 54 (4) (2011) 653-665.
S. Cen, X.R. Fu, G.H. Zhou M.J. Zhou and C.F. Li

The sensitivity problem to mesh distortion and the low accuracy problem of the stress solutions are two inherent difficulties in the finite element method. By applying the fundamental analytical solutions (in global Cartesian coordinates) to the Airy stress function φ of the anisotropic materials, 8- and 12- node plane quadrilateral hybrid stress-function (HS-F) elements are successfully developed based on the principle of the minimum complementary energy. Numerical results show that the present new elements exhibit much better and more robust performance in both displacement and stress solutions than those obtained from other models. They can still perform very well even when the element shapes degenerate into a triangle and a concave quadrangle. It is also demonstrated that the proposed construction procedure is an effective way for developing shape-free finite element models which can completely overcome the sensitivity problem to mesh distortion and can produce highly accurate stress solutions.

2010

A joint diagonalisation approach for linear stochastic systems
Computers and Structures, 88 (19-20) (2010) 1137-1148
C.F. Li, S. Adhikari, S. Cen, Y.T. Feng and D.R.J. Owen

In stochastic finite element problems the solution of a system of coupled linear random algebraic equations is needed. This problem in turn requires the calculation of the inverse of a random matrix. Over the past four decades several approximate analytical methods and simulation methods have been proposed for the solution of this problem in the context of probabilistic structural mechanics. In this paper we present a new solution method for stochastic linear equations. The proposed method is based on Neumann expansion and the recently developed joint diagonalisation solution strategy. Unlike the classical Neumann expansion, here only the inversion of a diagonal matrix is needed. Numerical examples are given to illustrate the use of the expressions derived in the paper.

A hybrid-stress element based on Hamilton principle
Acta Mechanica Sinica, 26 (4) (2010) 625-634
S. Cen, T. Zhang, C.F. Li, X.R. Fu and Y.Q. Long

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3ß here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-CReissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-CReissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3ß. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.

Example-based painting guided by color features
Visual Computer, 26 (6-8) (2010) 933-942
H. Huang, Y. Zang and C.F. Li

In this paper, by analyzing and learning the color features of the reference painting with a novel set of measures, an example-based approach is developed to transfer some key color features from the template to the source image. First, color features of a given template painting is analyzed in terms of hue distribution and the overall color tone. These features are then extracted and learned by the algorithm through an optimization scheme. Next, to ensure the spatial coherence of the final result, a segmentation based post processing is performed. Finally, a new color blending model, which avoids the dependence of edge detection and adjustment of inconvenient tune parameters, is developed to provide a flexible control for the accuracy of painting. Experimental results show that the new example-based painting system can produce paintings with specific color features of the template, and it can also be applied to changing color themes of art pieces, designing color styles of paintings/ real images, and specific color harmonization.

The elastic T-stress for slightly curved or kinked cracks
International Journal of Solids and Structures, 47 (2010) 1753-1763
D.F. Li, C.F. Li, H. Qing and J. Lu

This work presents a solution for the elastic T-stress at the tip of a slightly curved or kinked crack based on a perturbation approach. Compared to other exact or numerical solutions the present solution is accurate for considerable deviations from straightness. The T-stress variation as crack extends along a curved trajectory is subsequently examined. It is predicted that T-stress always keeps negative during crack extension when the crack has an initial negative T-stress. In the case of a positive T-stress and non-zero first and second stress intensity factors initially accompanying the crack, the T-stress is not positive with increasing the extension length until a threshold is exceeded. Based on directional stability criterion with respect to the sign of the T-stress, this result implies that for a straight crack with a positive T-stress, the crack extension path will not turn immediately and instead keep a stable growth until a critical length is reached. This prediction is consistent with experimental observations.

Axisymmetric vibration of singlewall carbon nanotubes in water
Physics Letters A, 373 (2010) 2467-2474
C. Y. Wang, C. F. Li and S. Adhikari

A double shell-Stokes flow model is developed to study the axisymmetric vibration of single-walled carbon nanotubes (SWCNTs) immerged in water. In contrast to macroscopic solid-liquid system, a submerged SWCNT is coupled with surrounding water via the van der Waals interaction. It is shown that this unique feature substantially reduces viscous damping of the axisymmetric radial, longitudinal and torsional vibrations and significantly up-shifts the frequency of the radial vibration of an SWCNT. The study offers a theoretical explanation for the experimental observation and molecular dynamics simulations available in particular cases, and provides an efficient modeling tool and useful guidance for the study of the general dynamic behavior of SWCNTs in a fluid.

A directed Monte Carlo solution of linear stochastic algebraic system of equations
Finite Elements in Analysis and Design, 46 (2010) 462-473
Y.T. Feng, C.F. Li and D.R.J. Owen

This paper proposes a modified Monte Carlo simulation method for the solution of a linear stochastic algebraic system of equations arising from the stochastic finite element modelling of linear elastic problems. The basic idea is to direct Monte Carlo samples along straight lines and then utilize their spatial proximity or order to provide high quality initial approximations in order to significantly accelerate the convergence of iterative solvers at each sample. The method, termed the directed Monte Carlo (DMC) simulation, is developed first for one random variable using the preconditioned conjugate gradient equipped with an initial approximation prediction scheme, and then extended to multiple Gaussian random variable cases by the adoption of a general hyper-spherical transformation. The eigen-poperties of the linear system are also briefly discussed to reveal the suitability of several preconditioning schemes for iterative solvers. Two numerical examples with up to around 6000 DOFs are provided to assess the performance of the proposed solution strategy and associated numerical techniques in terms of computational costs and solution accuracy.

Analytical trial function method for development of new 8-node plane element based on the variational principle containing Airy stress function
Engineering Computations, 24 (4) (2010) 442-463
X.R. Fu, S. Cen, C.F. Li and X.M. Chen

The purpose of this paper is to propose a novel and simple strategy for construction of hybrid-"stress function" plane element. First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8-node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. A new hybrid element stiffness matrix is obtained. The resulting 8-node plane element, denoted as analytical trial function (ATF-Q8), possesses excellent performance in numerical examples. Furthermore, some numerical defects, such as direction dependence and interpolation failure, are not found in present model. This paper presents a new strategy for developing finite element models exhibits advantages of both analytical and discrete method.

2009

Simulating Gaseous Fluids with Low and High Speeds
Computer Graphics Forum, 28 (7) (2009) 1845-1852
G. Yue, C.F. Li, S.M. Hu and B.A. Barsky

Gaseous fluids may move slowly, as smoke does, or at high speed, such as occurs with explosions. High-speed gas flow is always accompanied by low-speed gas flow, which produces rich visual details in the fluid motion. Realistic visualization involves a complex dynamic flow field with both low and high speed fluid behavior. In computer graphics, algorithms to simulate gaseous fluids address either the low speed case or the high speed case, but no algorithm handles both efficiently. With the aim of providing visually pleasing results, we present a hybrid algorithm that efficiently captures the essential physics of both low- and high-speed gaseous fluids. We model the low speed gaseous fluids by a grid approach and use a particle approach for the high speed gaseous fluids. In addition, we propose a physically sound method to connect the particle model to the grid model. By exploiting complementary strengths and avoiding weaknesses of the grid and particle approaches, we produce some animation examples and analyze their computational performance to demonstrate the effectiveness of the new hybrid method.

Dynamic behaviors of micro-tubules in cytosol
Journal of Biomechanics, 42 (9) (2009) 1270-1274.
C.Y. Wang, C.F. Li and S. Adhikari

Highly anisotropic micro-tubules (MTs) immersed in cytosol are a central part of the cytoskeleton in eukaryotic cells. The dynamic behaviors of an MT-cytosol system are of major interest in biomechanics community. Such a solid-fluid system is characterized by a Reynolds number of the order 10-3 and a slip ionic layer formed at the MT-cytosol interface. In view of these unique features, an orthotropic shell-Stokes flow model with a slip boundary condition has been developed to explore the distinctive dynamic behaviors of MTs in cytosol. Three types of motions have been identified, i.e., (a) undamped and damped torsional vibration, (b) damped longitudinal vibration, and (c) overdamped bending and radial motions. The exponentially decaying bending motion given by the present model is found to be in qualitative agreement with the existing experimental observation [Felgner et al., 1996. Flexural rigidity of microtubules measured with the use of optical tweezers, Journal of Cell Science 109, 509-516].

Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM-II)
International Journal for Numerical Methods in Engineering, 77 (8) (2009) 1172-1200
S. Cen, X.M. Chen, C.F. Li and X.R. Fu

A novel strategy for developing low-order membrane elements with analytical element stiffness matrices is proposed. First, some complete low-order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method (QACM-II). Then, these solutions are taken as the trial functions for developing new membrane elements. Thus, the interpolation formulae for displacement fields naturally possess second-order completeness in physical space (Cartesian coordinates). Finally, by introducing nodal conforming conditions, new 4-node and 5-node membrane elements with analytical element stiffness matrices are successfully constructed. The resulting models, denoted as QAC-ATF4 and QAC-ATF5, have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added. These two elements exhibit excellent performance in various bending problems with mesh distortion. It is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method, and the QACM-II is a powerful tool for constructing high-performance quadrilateral finite element models.

A Fourier-Karhunen-Loeve discretization scheme for stationary random material properties in SFEM
International Journal for Numerical Methods in Engineering, 73 (13) (2008) 1942-1965
C.F. Li, Y.T. Feng, D.R.J. Owen, D.F. Li and I.M. Davies

In order to overcome the computational difficulties in Karhunen-Loeve (K-L) expansions of stationary random material properties in stochastic finite element method (SFEM) analysis, a Fourier-Karhunen- Loeve (F-K-L) discretization scheme is developed in this paper, by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development. Three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical SFEM simulations. The proposed F-K-L discretization scheme exhibits a number of advantages over the widely used K-L expansion scheme based on FE meshes, including better computational efficiency in terms of memory and CPU time, convenient a priori error-control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes.

2008

A fast and accurate analysis of the interacting cracks in linear elastic solids
Internatioinal Journal of Fracture, 151 (2008) 169-185
D.F. Li, C.F. Li, S.Q. Shu, Z.X. Wang and J. Lu

This paper presents a fast and accurate solution for crack interaction problems in infinite- and half- plane solids. The newsolution is based on themethod of complex potentials developed by Muskhelishvili for the analysis of plane linear elasticity, and it is formulated through three steps. First, the problem is decomposed into a set of basic problems, and for each sub-problem, there is only one crack in the solid. Next, after a crack-dependent conformal mapping, the modified complex potentials associated with the sub-problems are expanded into Laurent's series with unknown coefficients, which in turn provides a mechanism to exactly implement in the form of Fourier series the boundary condition in each sub-problem. Finally, taking into account the crack interaction via a perturbation approach, an iterative algorithm based on fast Fourier transforms (FFT) is developed to solve the unknown Fourier coefficients, and the solution of the whole problem is readily obtained with the superposition of the complex potentials in each sub-problem. The performance of the proposed method is fully investigated by comparing with benchmark results in the literatures, and superb accuracy and efficiency is observed in all situations including patterns where cracks are closely spaced. Also, the new method is able to cope with interactions among a large number of cracks, and this capability is demonstrated by a calculation of effective moduli of an elastic solid with thousands of randomly-spaced cracks.

Discrete thermal element modelling of heat conduction in particle systems: Basic formulations
Journal of Computational Physics, 227 (2008) 5072-5089
Y.T. Feng, K. Han, C.F. Li and D.R.J. Owen

This paper proposes a novel numerical methodology, termed the discrete thermal element method (DTEM), for the effective modelling of heat conduction in systems comprising a large number of circular particles in 2D cases. Based on an existing analytical integral solution for the temperature distribution over a circular domain subjected to the Neumann boundary condition, a linear algebraic system of thermal conductivity equations for each particle is derived in terms of the average temperatures and the resultant fluxes at the contact zones with its neighboring particles. Thus, each particle is treated as an individual element with the number of (temperature) unknowns equal to the number of particles that it is in contact with. The element thermal conductivity matrix can be very effectively evaluated and is entirely dependent on the characteristics of the contact zones, including the contact positions and contact angles. This new element shares the same form and properties with its conventional thermal finite element counterpart. In particular, the whole solution procedure can follow exactly the same steps as those involved in the finite element analysis. Unlike finite elements or other modern numerical techniques, however, no discretization errors are involved in the discrete thermal elements. The modelling error mainly stems from the assumption made about the heat flux distribution within the contact zones. Based on some theoretical work, an enhanced version is suggested to improve the approximation. The numerical assessment against finite element results indicates that the basic version of DTEM can achieve a very reasonable solution accuracy, while the enhanced version further improves the accuracy to a high level. In addition, thermal resistance phenomena between the contact zones can be readily incorporated into the current modelling framework.

2006

Fourier representation of random media fields in stochastic finite element modelling
Engineering Computations, 23 (7) (2006) 794-817
C.F. Li, Y.T. Feng, D.R.J. Owen and I.M. Davies

To provide an explicit representation for wide-sense stationary stochastic fields which can be used in stochastic finite element modelling to describe random material properties. Design/methodology/approach - This method represents wide-sense stationary stochastic fields in terms of multiple Fourier series and a vector of mutually uncorrelated random variables, which are obtained by minimizing the mean-squared error of a characteristic equation and solving a standard algebraic eigenvalue problem. The result can be treated as a semi-analytic solution of the Karhunen-Loeve expansion. According to the Karhunen-Loeve theorem, a second-order stochastic field can be decomposed into a random part and a deterministic part. Owing to the harmonic essence of wide-sense stationary stochastic fields, the decomposition can be effectively obtained with the assistance of multiple Fourier series. The proposed explicit representation of wide-sense stationary stochastic fields is accurate, efficient and independent of the real shape of the random structure in consideration. Therefore, it can be readily applied in a variety of stochastic finite element formulations to describe random material properties. This paper discloses the connection between the spectral representation theory of wide-sense stationary stochastic fields and the Karhunen-Loeve theorem of general second-order stochastic fields, and obtains a Fourier-Karhunen-Loe`ve representation for the former stochastic fields.

Explicit solution to the stochastic system of linear algebraic equations (a1A1 + a2A2 + ... + amAm)x = b
Computer Methods in Applied Mechanics and Engineering, 195 (44-47) (2006) 6560-6576.
C.F. Li, Y.T. Feng and D.R.J. Owen

This paper presents a novel solution strategy to the stochastic system of linear algebraic equations (a1A1 + a2A2 + ... + amAm)x = b arising from stochastic finite element modelling in computational mechanics, in which ai (i = 1,. . . ,m) denote random variables, Ai (i = 1,. . . ,m) real symmetric deterministic matrices, b a deterministic/random vector and x the unknown random vector to be solved. The system is first decoupled by simultaneously diagonalizing all the matrices Ai via a similarity transformation, and then it is trivial to invert the sum of the diagonalized stochastic matrices to obtain the explicit solution of the stochastic equation system. Unless all the matrices Ai share exactly the same eigen-structure, the joint diagonalization can only be approximately achieved. Hence, the solution is approximate and corresponds to a particular average eigen-structure of the matrix family. Specifically, the classical Jacobi algorithm for the computation of eigenvalues of a single matrix is modified to accommodate multiple matrices and the resulting Jacobi-like joint diagonalization algorithm preserves the fundamental properties of the original version including its convergence and an explicit solution for the optimal Givens rotation angle. Three numerical examples are provided to illustrate the performance of the method.

SMB: Collision detection based on temporal coherence
Computer Methods in Applied Mechanics and Engineering, 195 (19-22) (2006) 2252-2269
C.F. Li, Y.T. Feng and D.R.J. Owen

The paper presents a novel collision detection algorithm, termed the sort moving boxes (SMB) for large number of moving 2D/3D objects which are represented by their axis-aligned bounding boxes (AABBs). The main feature of the algorithm is the full exploitation of the temporal coherence of the objects exhibited in a dynamic environment. In the algorithm, the AABBs are first projected to each Cartesian axis. The projected intervals on the axes are separately sorted by the diminishing increment sort (DIS) and further divided into subsections. By processing all the intervals within the subsections to check if they overlap, a complete contact list can be built. The SMB is a fast and robust collision detection algorithm, particularly for systems involving a large number of moving AABBs, and also supports for the dynamic insertion and deletion of objects. Its performance in terms of both expected total detection time and memory requirements is proportional to the total number of AABBs, N, and is not influenced by size differences of AABBs, the space size and packing density over a large range up to ten times difference. The only assumption made is that the sorted list at one time step will remain an almost sorted list at the next time step, which is valid for most applications whose movement and deformation of each AABB and the dynamic change of the total number N are approximately continuous.

2004

Actual morphing: a physics-based approach to blending
ACM Symposium on Solid Modelling and Applications, 309-314, Genoa, Italy, June 9-11, 2004
S.M. Hu, C.F. Li and H. Zhang

When two topologically identical shapes are blended, various possible transformation paths exist from the source shape to the target shape. Which one is the most plausible? Here we propose that the transformation process should obey a quasi-physical law. This paper combines morphing with deformation theory from continuum mechanics. By using strain energy, which reflects the magnitude of deformation, as an objective function, we convert the problem of path interpolation into an unconstrained optimization problem. To reduce the number of variables in the optimization we adopt shape functions, as used in the finite element method (FEM). A point-to-point correspondence between the source and target shapes is naturally established using these polynomial functions plus a distance map.

2002

Low-velocity impact-induced damage of continuous fiber-reinforced composite laminates. Part II. Verification and numerical investigation
Composites: Part A Applied Science and Manufacturing, 33 (8) (2002) 1063-1072.
C.F. Li, N. Hu, J.G. Cheng, H. Fukunaga and H. Sekine

In Part I of the current work [this issue], we have developed a numerical model for simulating the process of low-velocity impact damage in composite laminates using the finite element method (FEM). This FEM model based on the Mindlin plate element can describe various impact-induced damages and their mutual effects. Some new and effective techniques have also been put forward in that paper, which can significantly increase the computational efficiency. In the current paper, i.e. Part II of the two-part series on the study of impact of composites, we focus on the following two aspects: (a) verification of our numerical model through the comparison with other researchers' results; (b) investigation of the impact-induced damage in the laminated plates using the present numerical model. For the first aspect, some previous experimental data have been adopted for comparison to validate the present numerical model. For the second, we have mainly studied the effects on the impact damage in detail in such aspects as the size of target plate, the boundary conditions of target plate, impact velocity, impactor mass, etc. From these computations, the understanding of the low-velocity impact damage in laminates can be improved. q 2002 Elsevier Science Ltd. All rights reserved.

Low-velocity impact-induced damage of continuous fiber-reinforced composite laminates. Part I. An FEM numerical model
Composites: Part A Applied Science and Manufacturing, 33 (8) (2002) 1055-1062
C.F. Li, N. Hu, Y.J. Yin, H. Sekine and H Fukunaga

A numerical model for simulating the process of low-velocity impact damage in composite laminates using the finite element method is presented in this paper, i.e. Part I of this two part series on the study of impact. In this model, the 9-node Lagrangian element of the Mindlin plate with consideration of large deformation analysis is employed. To analyze the transient response of the laminated plates, a modified Newmark time integration algorithm previously proposed by the authors is adopted here. We also proved that the impact process between a rigid ball and laminated plates is a stiff system, therefore a kind of A(alpha) stable method has been advocated here to solve the motion equation of the rigid ball. Furthermore, various types of damages including delamination, matrix cracking and fiber breakage, etc. and their mutual influences are modeled and investigated in detail. To overcome the difficulty of numerical oscillation or instability in the analysis of the dynamic contact problem between delaminated layers using the traditional penalty methods, we have employed dynamic spring constraints to simulate the contact effect, which are added to the numerical model by a kind of continuous penalty function. Moreover, an effective technique to calculate the strain energy release rate based on the Mindlin plate model is proposed, which can attain high precision. Finally, some techniques of adaptive analyses have been realized for improving the computational efficiency. Based on this model, a program has been developed for numerically simulating the damage process of cross-ply fiber-reinforced carbon/epoxy composite laminates under lowvelocity impact load. In Part II, this numerical model will be verified by comparing with the experimental results. Also the impact damage will be investigated in detail using this numerical approach.

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# Unsupervised learning

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Unsupervised learning is a term used for Hebbian learning, associated to learning without a teacher, also known as self-organization and a method of modelling the probability density of inputs.[1]

The cluster analysis is a branch of machine learning that groups the data that has not been labelled, classified or categorized. Instead of responding to feedback, cluster analysis identifies commonalities in the data and reacts based on the presence or absence of such commonalities in each new piece of data.

A central application of unsupervised learning is in the field of density estimation in statistics,[2] though unsupervised learning encompasses many other domains involving summarizing and explaining data features. It could be contrasted with supervised learning by saying that whereas supervised learning intends to infer a conditional probability distribution ${\textstyle p_{X}(x\,|\,y)}$ conditioned on the label ${\textstyle y}$ of input data; unsupervised learning intends to infer a priori probability distribution ${\textstyle p_{X}(x)}$.

## Approaches

Compared to supervised learning where training data is labeled with the appropriate classifications, models using unsupervised learning must learn relationships between elements in a data set and classify the raw data without "help." This hunt for relationships can take many different algorithmic forms, but all models have the same goal of mimicking human logic by searching for indirect hidden structures, patterns or features to analyze new data.[3]

Some of the most common algorithms used in unsupervised learning include:

## Neural networks

The classical example of unsupervised learning in the study of neural networks is Donald Hebb's principle, that is, neurons that fire together wire together. In Hebbian learning, the connection is reinforced irrespective of an error, but is exclusively a function of the coincidence between action potentials between the two neurons. A similar version that modifies synaptic weights takes into account the time between the action potentials (spike-timing-dependent plasticity or STDP). Hebbian Learning has been hypothesized to underlie a range of cognitive functions, such as pattern recognition and experiential learning.

Among neural network models, the self-organizing map (SOM) and adaptive resonance theory (ART) are commonly used in unsupervised learning algorithms. The SOM is a topographic organization in which nearby locations in the map represent inputs with similar properties. The ART model allows the number of clusters to vary with problem size and lets the user control the degree of similarity between members of the same clusters by means of a user-defined constant called the vigilance parameter. ART networks are used for many pattern recognition tasks, such as automatic target recognition and seismic signal processing.[5]

## Method of moments

One of the statistical approaches for unsupervised learning is the method of moments. In the method of moments, the unknown parameters (of interest) in the model are related to the moments of one or more random variables, and thus, these unknown parameters can be estimated given the moments. The moments are usually estimated from samples empirically. The basic moments are first and second order moments. For a random vector, the first order moment is the mean vector, and the second order moment is the covariance matrix (when the mean is zero). Higher order moments are usually represented using tensors which are the generalization of matrices to higher orders as multi-dimensional arrays.

In particular, the method of moments is shown to be effective in learning the parameters of latent variable models.[6] Latent variable models are statistical models where in addition to the observed variables, a set of latent variables also exists which is not observed. A highly practical example of latent variable models in machine learning is the topic modeling which is a statistical model for generating the words (observed variables) in the document based on the topic (latent variable) of the document. In the topic modeling, the words in the document are generated according to different statistical parameters when the topic of the document is changed. It is shown that method of moments (tensor decomposition techniques) consistently recover the parameters of a large class of latent variable models under some assumptions.[6]

The Expectation–maximization algorithm (EM) is also one of the most practical methods for learning latent variable models. However, it can get stuck in local optima, and it is not guaranteed that the algorithm will converge to the true unknown parameters of the model. In contrast, for the method of moments, the global convergence is guaranteed under some conditions.[6]

## Notes

1. ^ Hinton, Jeffrey; Sejnowski, Terrence (1999). Unsupervised Learning: Foundations of Neural Computation. MIT Press. ISBN 978-0262581684.
2. ^ Jordan, Michael I.; Bishop, Christopher M. (2004). "Neural Networks". In Allen B. Tucker (ed.). Computer Science Handbook, Second Edition (Section VII: Intelligent Systems). Boca Raton, Florida: Chapman & Hall/CRC Press LLC. ISBN 1-58488-360-X.
3. ^ "Build with AI | DeepAI". DeepAI. Retrieved 2018-09-30.
4. ^ Hastie, Trevor, Robert Tibshirani, Friedman, Jerome (2009). The Elements of Statistical Learning: Data mining, Inference, and Prediction. New York: Springer. pp. 485–586. ISBN 978-0-387-84857-0.CS1 maint: Multiple names: authors list (link)
5. ^ Carpenter, G.A. & Grossberg, S. (1988). "The ART of adaptive pattern recognition by a self-organizing neural network" (PDF). Computer. 21: 77–88. doi:10.1109/2.33.
6. ^ a b c Anandkumar, Animashree; Ge, Rong; Hsu, Daniel; Kakade, Sham; Telgarsky, Matus (2014). "Tensor Decompositions for Learning Latent Variable Models" (PDF). Journal of Machine Learning Research (JMLR). 15: 2773–2832.


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### January 2008

Today, rather than posting in my Algebraic Geometry from the Beginning series, I’m just going to do a quick post on a few things that are going on mathematically. First off, by clicking through Kung Fu Physics’s Super Holiday Post because it linked here, I found the Academic Reader. It’s more aimed at physics than math, but it still seems nifty, and looks like it might be a bit more suited to my paper organizing and reading needs than Google Reader (which I do make massive use of).

On the topic of papers, I’m about to start reading a couple of papers about Non-Reductive Geometric Invariant Theory. One of them popped up on the Arxiv today, on Quotients by non-reductive algebraic group actions by Kirwan. Fortunately, the abstract has a link to an older paper, Towards non-reductive geometric invariant theory by Doran and Kirwan last year. I’m excited about these because I’m organizing the Graduate Student Algebra Seminar here at Penn this semester, and we decided that our topic was going to be algebraic groups and invariant theory. I’m giving a couple of the early talks focusing on invariants of finite groups, and I’ll probably post my talks here after I give them.

Additionally, I’m about to start reading Meet homological mirror symmetry by Ballard because I’ve been wanting to figure out what this stuff is, and I think now I’m getting to the place where I have a chance at understanding it, being in my second semester of Complex Algebraic Geometry, which is focusing on open problems (Geometric Langlands, Hodge Conjecture, and Mirror Symmetry seem to be the ones we’re focusing on), I’m studying Hartshorne and doing the problems in it whenever I have spare time to prep for my orals in the Fall, and I’m doing a reading course on Deformation Theory, starting with Manetti’s Lectures on Deformations of Complex Manifolds. Overall, I’ve got quite a semester in front of me, and I’ll keep updating here, well…whenever it crosses my mind or whenever I want to post but don’t feel like doing anything technical.

We’ve got sheaves now, so naturally we move on to morphisms between them. We begin by fixing $X$ a topological space and take $\mathscr{F},\mathscr{G}$ to be two presheaves on $X$. A morphism of presheaves is an abelian group homomorphism $\phi(U):\mathscr{F}(U)\to\mathscr{G}(U)$ for each $U\subset X$ open, that satisfies an extra condition.

Second in our survey of some technical tools are sheaves. Sheaves have a reputation for being terrifying and technical and some people I know have trouble seeing their purpose at all. In my opinion, they’re very nice, convenient ways to organize information. Sheaves do require a bit of topology to use, and I recommend John Armstrong’s exposition as a basic reference. I talked a bit about the topology relevant in algebraic geometry (which is rather different than that in most point set topology settings, note that we make use of the concept of irreducibility quite often) but the basics and the notion of connectedness are both still rather important.

Now that I’m settling into the semester, it’s time that I get back to rambling semi-coherently about algebraic geometry. I do, however, now have a somewhat more focused goal, and though it’s still quite a ways off, it has reminded me of important machinery that needs introduction. So for a bit, I’ll be talking more about algebra and some other tools of the trade and putting the geometry itself on the back burner for a bit (which isn’t to say that I’m abandoning it completely).

The semester is starting up again now, so my schedule is a bit irregular. Sadly, I won’t be able to keep up the pace I set over break. However, I should still be putting out a couple of posts a week. So without further ado, tangent spaces. We’ve looked at tangent spaces before, and used them to define what it means for a point to be smooth versus singular. Today, we’re going to start back at the beginning of this stuff and justify the definitions a little.

Back in the second post, we defined projective space to be the collection of lines through the origin in affine space. A natural generalization is to look at $r$-planes through the origin in affine space. At first glance, we might think that these spaces are legitimate generalizations of projective space, and that it might be valuable to do algebraic geometry on them.

Today, we talk about the Segre Embedding. This will let us say a bit more about the products of varieties that were mentioned before.

Now we’ve got some basic theory under our belts. Time to get a handle on specific objects. Nothing gives better intuition into algebraic geometry than working out a bunch of examples and seeing how they fit together. We’ll begin with the Veronese Embedding, which will give us a bunch of varieties that are isomorphic to projective space.

We’ve now spoken about all sorts of invariants, the most important ones being derived from the Hilbert Polynomial, we’ve spoken about Bezout’s Theorem, and we’ve developed a nice bit of classical algebraic geometry. We’ve also mentioned algebraic groups, and it might be a good idea to refresh this stuff, because now we’ll return to it, and build up some more examples.

Now we’re ready for our first big geometric theorem. We’ll prove Bezout’s Theorem as a special case of another, more general theorem.

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 Math Central Quandaries & Queries
 Question from Cody: if a train travels 400 meters in 30 seconds how many kilometers per hr is it traveling

Ho Cody,

One kilometer is 1000 meters so

$\mbox{400 meters is } \frac{400}{1000} \mbox{ kilometers.}$

One hour is 60 minutes and one minute is 60 seconds so one hour is $60 \times 60$ seconds or

$\mbox{one second is} \frac{1}{60 \times 60} \mbox{ hours.}$

Hence the train travels

$\frac{400}{1000} \mbox{ kilometers in }30 \times \frac{1}{60 \times 60} \mbox{ hours.}$

How far does it travel in one hour?

Penny


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Documentation

Contents

Estimate radius of ellipsoid planet at geocentric latitude

Library

Flight Parameters

Description

The Radius at Geocentric Latitude block estimates the radius $\left({r}_{s}\right)$ of an ellipsoid planet at a particular geocentric latitude $\left({\lambda }_{s}\right)$.

The following equation estimates the ellipsoid radius $\left({r}_{s}\right)$ using flattening $\left(\overline{f}\right)$, geocentric latitude $\left({\overline{\lambda }}_{s}\right)$, and equatorial radius $\left(\overline{R}\right)$.

${r}_{s}=\sqrt{\frac{{R}^{2}}{1+\left[1/{\left(1-f\right)}^{2}-1\right]{\mathrm{sin}}^{2}{\lambda }_{s}}}$

Dialog Box

Units

Specifies the parameter and output units:

Metric (MKS)MetersMeters
EnglishFeetFeet

This option is only available when Planet model is set to Earth (WGS84).

Planet model

Specifies the planet model to use:

Custom

Earth (WGS84)

Flattening

Specifies the flattening of the planet. This option is only available with Planet model set to Custom.

Specifies the radius of the planet at its equator. This option is only available with Planet model set to Custom.

Inputs and Outputs

InputDimension TypeDescription

First

Contains the geocentric latitude, in degrees.

OutputDimension TypeDescription

First

Contains the radius of planet at geocentric latitude, in the same as the units as flattening.

References

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.

Zipfel, P. H., Modeling and Simulation of Aerospace Vehicle Dynamics, AIAA Education Series, Reston, Virginia, 2000.