Finite Element Method System Research Articles

Linear viscoelastic transversely isotropic model based on the spectral decomposition of elasticity tensors

The linear viscoelasticity is still a useful model in the engineering for studying the behavior of materials loaded with different loading rates (frequencies). Certain types of materials reveal also an anisotropic behavior: fiber reinforced composites, asphalt concrete mixtures, or wood, to name a few. In general, researchers try to identify experimentally the dependence of engineering constants like: directional Young’s moduli and Poisson’s ratios on loading velocity by means of creep or harmonic oscillatory tests. This approach is appealing from the experimental point of view. However, from the modeling perspective, this is not the case. The engineering constants emerge in nonlinear manner in the relationship between the strain and stress via fourth order stiffness tensor components. This is especially true in higher order anisotropies, yet even in isotropy Poisson’s ratio appears nonlinearly in the stiffness tensor. Several models for the linear viscoelasticity of anisotropic materials already exist in the literature that try to tackle this issue. In this paper, we propose a linear viscoelasticity model for anisotropic materials based on the spectral decomposition of the stiffness tensor. The proposed model offers several advantages: the natural choice of the stiffness tensor eigenvalues as time-dependent variables, state variables with clear interpretation as creep strains, and reduced burden of storage utilizing the orthogonality of the eigenspaces. We implemented the model in the finite-element method system AceGen/AceFEM, and calibrated parameters of the model with the experimental data available in the literature, thus proving the adequacy of the proposed model to describe anisotropic viscoelastic materials.

Advanced Simulation-Inserted Optimization Using Combined Quasi-Newton Method With Lagrangian Method for EM-Based Design Optimization

Traditional electromagnetic (EM)-based design optimizations typically treat the EM simulation as a black-box and use optimization algorithms to externally drive EM simulations iteratively. Recently, a novel EM-based design optimization technique, named as simulation-inserted optimization (SIO) in this article, has been developed by opening the black-box and inserting the optimization algorithm into the internal of EM simulation to achieve EM field solutions and the optimal values of design variables simultaneously. This article proposes an advanced SIO algorithm using combined quasi-Newton method with Lagrangian method. In the proposed algorithm, the finite element method (FEM) is used as the EM simulation method, and the Lagrangian method is incorporated to internally integrate EM simulation with design optimization. New formulations based on quasi-Newton method are derived to approximate the computationally intensive lower upper (LU) decomposition of the FEM system matrix in the current optimization iteration by that in the previous optimization iteration to significantly reduce the number of LU decompositions. The proposed SIO technique further derives novel optimization update formulations by introducing the line search algorithm to increase the robustness of optimization. Using the proposed SIO algorithm, only a few initial LU decompositions of FEM system matrices need to be calculated instead of repetitively calculating a large number of LU decompositions of FEM system matrices during optimization, consequently speeding up the overall optimization process. The proposed technique is demonstrated by two application examples of EM-based design optimization of microwave components.

An MOR Algorithm Based on the Immittance Zero and Pole Eigenvectors for Fast FEM Simulations of Two-Port Microwave Structures

The aim of this article is to present a novel model-order reduction (MOR) algorithm for fast finite-element frequency-domain simulations of microwave two-port structures. The projection basis used to construct the reduced-order model (ROM) comprises two sets: singular vectors and regular vectors. The first set is composed of the eigenvectors associated with the poles of the finite-element method (FEM) state-space system, while the second one is made up from the eigenvectors corresponding to the zeros of the diagonal elements of the matrix-valued immittance transfer function. Importantly, just one LU factorization of the FEM system is required to construct the projection basis during the reduction process, due to the application of a new formulation based on the Schur complement. The sets of eigenvectors that are used in the basis are independent of one another, which makes the new technique better suited for parallel computing compared with previously developed methods, which are sequential in nature. The reliability and accuracy of the proposed scheme are compared with that of the standard MOR technique, namely, the reduced-basis method (RBM), and verified through the analysis of three microwave structures: an eighth-order dual-mode waveguide filter, a dielectric resonator filter, and a folded waveguide filter.

A deep learning-based composite design strategy for efficient selection of material and layup sequences from a given database

A machine learning-assisted composite design framework is established in this paper as an effective and efficient way to find feasible or optimal selections of fiber materials and layup stacking orientations to meet the mechanical and non-mechanical requirements. With a given material database and conventional guidelines for layup design, a generative machine learning model using the physical-informed Conditional Generative Adversarial Networks (CGAN) is developed to provide solutions suitable for various design boundaries such as target strength, maximum deformation, minimum thickness and lowest cost. In order to import the physical constraints in the form of inequalities into the CGAN model, a customized loss along with the traditional loss function of CGAN are applied to the generator for generating outputs that are more favorable to our design requirements as well as conforming to our input data pattern. The networks are trained by using a total of 3000 Finite Element Method (FEM) simulation data from randomly generated composite configurations. Following an evaluation of the training performance of the model, this design strategy is applied to three benchmark problems of a composite tube under different loading conditions and design constraints. The results show that the CGAN model can successfully provide a large number of FRP configurations that fit into the scope of mechanical requirements. In the meanwhile, this integrated CGAN model and FEM system is easily extendable to other composite inverse design scenarios.

Finite element modelling of fully-coupled flow/fiber-orientation effects in polymer composite deposition additive manufacturing nozzle-extrudate flow

This paper presents a Finite Element Method (FEM) based algorithm to simulate the mutually dependent effects between non-Newtonian polymer melt flow and fiber orientation in Polymer Composite Deposition Additive Manufacturing (PCDAM). The computational approach is employed to analyze an axisymmetric flow with a free surface that defines the melt extrudate. The non-Newtonian power law model is applied to quantify the shear thinning rheological behavior of the polymer flow. The fiber orientation state of the composite melt is simulated using the Advani-Tucker orientation tensor approach, and a streamline-based remeshing technique is applied to compute die swell of the free extrudate. The non-linear FEM system is solved with the Newton Raphson iteration method. Computed results are compared to those obtained using a flow-fiber one way coupling simulation, where it is shown that the magnitude of flow field along the direction of extrusion, as well as the fiber alignment in the flow direction, both increase when full coupling is used. In addition, the extrudate swell ratio solved by the fully-coupled scheme reduces by roughly a factor of ~2× as compared to the weakly-coupled flow solution.

Stable high-order absorbing boundary condition based on new continued fraction for scalar wave propagation in unbounded multilayer media

A long-time stable high-order absorbing boundary condition (ABC) is developed for the finite element simulation of time-dependent scalar wave propagation in unbounded multilayer media that is a multilayer waveguide. The ABC is obtained based on a new continued fraction (CF) expansion of the frequency-domain dynamic stiffness matrix on the artificial boundary of a truncated infinite domain. The dynamic stiffness in scalar CF form is first proposed in the modal space for a single-layered medium and is subsequently extended to a matrix CF form for multilayer media. The new CF converges to dynamic stiffness over the whole frequency range as its order increases for the single-layered medium, and has high accuracy for the multilayer media. The CF-based high-order ABC is therefore accurate and stable in the time domain. After being coupled seamlessly with finite element method (FEM), instability phenomena are observed during long-time computation. In order to eliminate the instability of the coupled ABC and FEM system, a straightforward and effective method is proposed by introducing damping proportional to the stiffness matrix of spatially discretised finite domain. Numerical examples demonstrate the superior properties of the proposed ABC with high accuracy and long-time stability as well as remarkable coupling with FEM.

Mesh Partitioning and Efficient Equation Solving Techniques by Distributed Finite Element Methods: A Survey

The mesh partitioning in parallel Finite Element Method (FEM) is an NP-hard problem. During the past few decades, several heuristic approaches have been proposed to address this problem. In addition to mesh distribution, solving a large set of algebraic equations also significantly contributes to the performance of a parallel solution. A number of efficient equation solving techniques are developed which exploit inherent properties of large coefficient matrices (for instance, symmetry and positive definiteness). In the present study, the performance of a distributed FEM system on the basis of the mesh partitioning approaches and equation solvers is discussed. The work contributes towards: (i) categorizing mesh partitioning methods, (ii) examining implementation variations in linear and nonlinear solution of equations, and (iii) exploring the impact of mesh partitioning and an equation solver on the performance of a distributed FEM system.

Advanced Metal Forming Simulations Coupled with Neutron Beam Technology

A finite-element method (FEM) system has been developed, which enables us to perform structural analyses by using measured data obtained by 3D measurement system such as X-ray or neutron beam computed tomography (CT). This system allows us to use more realistic model of industrial products than models generated by conventional method based on CAD. On the other hand, the FEM solver included in this system employs the elastoplastic material model and the updated Lagrangian rate formulation to describe finite deformation problems. With these features, the solver is applicable to a variety of manufacturing processes such as sheet metal forming. In this study, Cockcroft and Latham's ductile fracture criterion was introduced into this code to simulate shear-cutting processes. Several experiments with ductile fracture propagation were carried out and the results were compared with those obtained by FE simulations made by STAMP3D. The robustness and accuracy of the developed FE code for the simulations of metal forming processes with ductile fracture such as shearing process were proved in this study. The above mentioned FEM system is planned to be coupled with crystal plasticity analysis solver so that the results can be verified by comparing the crystal orientation distribution obtained by neutron sources.

The Development and Simulation of the 3D FEM Software ---- Alpha System

Aimed at the Chinese FEM (Finite Element Method) software market was occupied by foreign company, the article introduced the FEM software----Alpha System, which was developed by us. Alpha System could deal with heat transfer and structure static problems. It had 6 modules such as project management, geometric modeling, meshing, loading, solving, and visualization. Some of the key technologies were afforded to solve the difficulties on achieving the software. At last, a part of car rear axle was simulated by Alpha and Ansys. The contrast analysis of the result proved the accuracy of Alpha System. The article gives the reference on how to design the FEM System.

Fast Magnetic Flux Leakage Signal Inversion for the Reconstruction of Arbitrary Defect Profiles in Steel Using Finite Elements

This paper proposes a fast and effective method for reconstructing arbitrary defect profiles in steel plates from magnetic flux leakage (MFL) measurements widely used in nondestructive testing (NDT) of oil storage tanks and pipelines. The inverse reconstruction problem is formulated based on a nonlinear forward model using the finite element method (FEM) and is implemented in 2-D. A Gauss-Newton optimization is applied to reconstruct the defect geometry, using efficiently calculated Jacobian information directly derived from the FEM system matrix.

Research on the mechanism of flash line defect in coining

Flash line is one of the most important surface defects in coining process of silver commemorative coin. Due to the deficiency of study on the defect mechanism, it takes a long time to eliminate the defect through die tryouts which bring significant instabilities and development costs in coin manufacture. In the present investigation, the mechanism of flash line is studied by the elimination solutions and metal flow analysis using the finite element method (FEM). The radial components of friction in die–workpiece interface are considered as the main reasons of the defect. A modified friction model considering the influence of the sliding velocity in the die–workpiece interface is employed to describe the frictional behaviors in coining more properly. A novel radial friction work (RFW) model is proposed to predict the tendency of flash line and employed to a dynamic explicit FEM system for coining-COINFORM. The comparisons of the numerical and experimental results of a typical silver coin demonstrate that the RFW model is valid in the prediction of flash line.

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and ApplicationsThis paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.

A Time-Harmonic Three-Dimensional Vector Boundary Element Model for Electromechanical Devices

In present practice, the most effective way to solve the large electromagnetic (EM) boundary value problems typical in electromechanical device analysis has been with the finite element method (FEM). The sparse, symmetric, and banded structure of FEM system matrices reduces the memory requirements and facilitates several fast and efficient solution algorithms. An alternative, boundary element methods (BEM), is more computationally intensive. Recently, however, fast and efficient solver codes have been developed for BEM solutions of EM scattering problems. These, if effectively implemented in electromechanical device models, can make BEM a more feasible alternative for this purpose than previously. To generate a deeper understanding of this alternative formulation in the context of electromechanics problems, a time-harmonic 3-D vector BEM model for electromechanical devices is presented that is formulated in terms of the field variables and is capable of modeling multiple separated homogeneous regions with or without eddy currents. Extensions to electric machine modeling are given, and the model is assessed using experimental data.

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

AbstractIn this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.

High-accuracy schemes of the finite element method for systems of degenerate elliptic equations on an interval

In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop high-accuracy schemes based on multiplicative singularity extraction. We prove theorems about the smoothness of a solution. Based on these theorems we estimate the error of the proposed method.

Parameterized Model Order Reduction Techniques for FEM Based Full Wave Analysis

As operating frequencies increase full wave methods such as the finite element method (FEM) become necessary for the analysis of high-frequency circuit structures. Such techniques result in very large systems of equations, and model order reduction (MOR) was proven to be very effective in combating such increased complexity. Using traditional MOR, one has to generate a new reduced model each time a design parameter is modified, thus significantly reducing the CPU efficiency. In this paper, a multidimensional Krylov subspace method is proposed to perform reduction directly on the vector wave equation based FEM system and to generate parametric reduced order models that are valid over the desired parameter range without the need to redo the reduction. In order to accomplish this, second-order Arnoldi methods are extended to include design parameters such as material properties, and geometrical parameters in the reduced order model. In addition, multidimensional moment matching technique is used to address the Krylov incompatibility of FEM problems which include arbitrary frequency dependence in the system. This technique results in significant CPU savings and enables applications such as optimization and design space exploration.

Rigorous, Auxiliary Variable-Based Implementation of a Second-Order ABC for the Vector FEM

The finite element method (FEM) is commonly used for electromagnetic radiation and scattering analysis. When an infinite, free space exterior domain needs to be incorporated into the method, a radiation boundary condition must be enforced. An approach which has received considerable attention, is to employ approximate conditions, known as absorbing boundary conditions (ABCs), that preserve the sparsity of the original FEM system upon discretization. In the case of time-harmonic analysis based on the vector wave equation in three dimensions, the symmetric, spherical Bayliss-Turkel-type ABCs of first- and second-orders are well-established. The second-order version is expected to be more accurate, however when using the standard curl-conforming approach to FEM discretization, an implementation difficulty is encountered, relating to successive derivatives being required of the nonconforming field components. This issue is addressed here by introducing a scheme where the nonconforming first-order derivatives are projected onto a suitably conforming auxiliary field, of which another derivative can then be taken instead. Additional computational costs are minimal and the scheme retains the symmetry of the standard formulation. Numerical results demonstrate the superior performance of the rigorously implemented second-order ABC over its first-order counterpart

Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis

A method of reducing the number of degrees of freedom and the overall computing times in finite element method (FEM) has been devised. The technique is valid for linear problems and arbitrary temporal variation of boundary conditions. At the first stage of the method standard FEM time stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. Simple matrix manipulations applied to the original FEM system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original FEM formulation. Using the concept of modal analysis the set is then solved analytically. Treatment of non-homogeneous initial conditions, time-dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed. Several numerical examples are included for validation of the approach. Copyright © 2004 John Wiley & Sons, Ltd.

Implementation of direct and iterative methods for a class of Helmholtz wave problems

In this article, the applicability and the performance of two direct (DirCSR and DirSkyline) and one iterative (IterCG_CSR) methods for solving large linear systems of equations are analyzed. Alternative ways of storing the coefficient matrix K generated by the application of the finite element method (FEM) to the Helmholtz equation for wave propagation in coastal regions are considered. The two direct methods are based on a Crout variant, the LDL t factorization, but they are implemented with two different sparse storage modes, one being the standard skyline, and the other being the compressed sparse row. The preconditioner of the conjugate gradients iterative method is the above DirCSR, with a minor parameter adjustment. These methods are compared in terms of CPU time, size of the matrices and number of iterations, for the same systems of equations with complex symmetric and indefinite matrices. Their behavior in low resolution grids is also investigated because this is a critical issue in practical applications. All these methods were implemented in a wave propagation model (DREAMS) for harbours and sheltered zones, which is used to generate the FEM systems of equations to be solved.

Implementation of direct and iterative methods for a class of Helmholtz wave problems

In this article, the applicability and the performance of two direct (DirCSR and DirSkyline) and one iterative (IterCG_CSR) methods for solving large linear systems of equations are analyzed. Alternative ways of storing the coefficient matrix K generated by the application of the finite element method (FEM) to the Helmholtz equation for wave propagation in coastal regions are considered. The two direct methods are based on a Crout variant, the LDLt factorization, but they are implemented with two different sparse storage modes, one being the standard skyline, and the other being the compressed sparse row. The preconditioner of the conjugate gradients iterative method is the above DirCSR, with a minor parameter adjustment. These methods are compared in terms of CPU time, size of the matrices and number of iterations, for the same systems of equations with complex symmetric and indefinite matrices. Their behavior in low resolution grids is also investigated because this is a critical issue in practical applications. All these methods were implemented in a wave propagation model (DREAMS) for harbours and sheltered zones, which is used to generate the FEM systems of equations to be solved.