Maintaining large time steps in explicit finite element simulations using shape matching

We present a novel hybrid method to allow large time steps in explicit integrations for the simulation of deformable objects. In explicit integration schemes, the time step is typically limited by the size and the shape of the discretization elements as well as by the material parameters. We propose a two-step strategy to enable large time steps for meshes with elements potentially destabilizing the integration. First, the necessary time step for a stable computation is identified per element using modal analysis. This allows determining which elements have to be handled specially given a desired simulation time step. The identified critical elements are treated by a geometric deformation model, while the remaining ones are simulated with a standard deformation model (in our case, a corotational linear Finite Element Method). In order to achieve a valid deformation behavior, we propose a strategy to determine appropriate parameters for the geometric model. Our hybrid method allows taking much larger time steps than using an explicit Finite Element Method alone. The total computational costs per second are significantly lowered. The proposed scheme is especially useful for simulations requiring interactive mesh updates, such as for instance cutting in surgical simulations.     

An extended finite element method for modeling crack growth with frictional contact

A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.     

Surface Representation with Closest Point Projection in the X-FEM

Mathematical Models and Computer Simulations - At present, the eXtended Finite Element Method (X-FEM) is a common generalization of the classical finite element method for solving problems of...     

The ALE/Lagrangian Particle Finite Element Method: A new approach to computation of free-surface flows and fluid-object interactions

The Particle Finite Element Method (PFEM) is a well established numerical method [Aubry R, Idelsohn SR, Oñate E, Particle finite element method in fluid mechanics including thermal convection-diffusion, Comput Struct 2004;83:1459-75; Idelsohn S, Oñate E, Del Pin F, A Lagrangian meshless finite element method applied to fluid-structure interaction problems, Comput Struct 2003;81:655-71; Idelsohn SR, Oñate E, Del Pin F, The particle finite element method a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Int J Num Methods Eng 2004;61:964-84] where critical parts of the continuum are discretized into particles. The nodes treated as particles transport their momentum and physical properties in a Lagrangian way while the rest of the nodes may move in an Arbitrary Lagrangian-Eulerian (ALE) frame. In order to solve the governing equations that represent the continuum, the particles are connected by means of a Delaunay Triangulation [Idelsohn SR, Oñate E, Calvo N, Del Pin F, The meshless finite element method, Int J Num Methods Eng 2003;58(4)]. The resulting partition is a mesh where the Finite Element Method is applied to solve the equations of motion. The application of a fully Lagrangian formulation on the particles provides a natural and simple way to track free surfaces as well as to compute contacts in an accurate and robust fashion. Furthermore, the usage of an ALE formulation allows large mesh deformation with larger time steps than the full Lagrangian scheme.     

An algorithm for automatic 2D quadrilateral mesh generation with line constraints

Finite element method (FEM) is a fundamental numerical analysis technique widely used in engineering applications. Although state-of-the-art hardware has reduced the solving time, which accounts for a small portion of the overall FEM analysis time, the relative time needed to build mesh models has been increasing. In particular, mesh models that must model stiffeners, those features that are attached to the plate in a ship structure, are imposed with line constraints and other constraints such as holes. To automatically generate a 2D quadrilateral mesh with the line constraints, an extended algorithm to handle line constraints is proposed based on the constrained Delaunay triangulation and Q-Morph algorithm. The performance of the proposed algorithm is evaluated, and numerical results of our proposed algorithm are presented.     

Fast and robust level set update for 3D non-planar X-FEM crack propagation modelling

In the X-FEM framework, the need to represent a discontinuity independently of the structural mesh relies on the level set technique. Hence crack propagation can be simulated by an update of two distinct level sets, the evolution of which is described by differential equations. The aim of this paper is to analyse the resolution of these equations in order to formulate a robust and fast numerical process allowing 3D crack propagation simulations even in presence of high kink angles occurring in mixed mode propagation. The numerical integration is accomplished by means of a robust finite difference upwind scheme applied to an auxiliary regular grid. An alternative level set update equation and a fast localisation of the integration domain, specifically developed for crack propagation problems, are formulated and proposed in the paper in order to gain in stability, robustness and performance.     

A methodology for adaptive mesh refinement in optimum shape design problems

This work presents a methodology based on the use of adaptive mesh refinement (AMR) techniques in the context of shape optimization problems analyzed by the Finite Element Method (FEM). A suitable and very general technique for the parametrization of the optimization problem using B-splines to define the boundary is first presented. Then, mesh generation using the advancing front method, the error estimation and the mesh refinement criteria are dealt with in the context of a shape optimization problems. In particular, the sensitivities of the different ingredients ruling the problem (B-splines, finite element mesh, design behaviour, and error estimator) are studied in detail. The sensitivities of the finite element mesh coordinates and the error estimator allow their projection from one design to the next, giving an “a priori knowledge” of the error distribution on the new design. This allows to build up a finite element mesh for the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked out with some 2D examples.     

Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method

This paper deals with some aspects related to three-dimensional numerical modelling of reinforced concrete structures using the Finite Element Method (FEM). Some subjects such as the solution technique of the non-linear equilibrium equations and the constitutive model for concrete and reinforcement steel are emphasised and commented. A robust method for the evaluation of the intersecting points of the embedded reinforcement bars into the three-dimensional finite element mesh is also presented. The main advantages of the Generalised Displacement Control Method with the Generalised Displacement Parameter to improve the response of the concrete and reinforced concrete analyses are highlighted. Finally, a series of numerical examples related to the above-mentioned aspects are presented.     

Structural shape optimization using Cartesian grids and automatic <Emphasis Type="Italic">h</Emphasis>-adaptive mesh projection

We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of the matrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h-adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process.     

基于ANSYS的铝电解槽电场数值仿真研究

铝电解槽的电场作为形成各物理场的基础,其分布好坏直接影响电解槽的生产。采用基于ANSYS的仿真分析方法研究了铝电解槽的电场分布。采用有限元法,建立了铝电解槽电场的数学模型,并利用有限元仿真分析软件ANSYS建立电解槽三维电场有限元模型,计算了铝电解槽各导电部分的电场分布,总结了分布规律,为电解槽多物理场优化提供基础。其仿真结果与实测结果相吻合,验证了方法的正确性。     

二维不可压缩Navier-Stokes方程的并行谱有限元法求解

针对不可压缩Navier-Stokes （N-S）方程求解过程中的有限元法存在计算网格量大、收敛速度慢的缺点,提出了基于面积坐标的三角网格剖分谱有限元法（TSFEM）并进一步给出了利用OpenMP对其并行化的方法。该算法结合谱方法和有限元法思想,选取具有无限光滑特性的指数函数取代传统有限元法中的多项式函数作为基函数,能够有效减少计算网格数量,提高算法的精度和收敛速度;利用面积坐标便于三角形单元计算的特点,选取三角单元作为计算单元,增强了适用性;在顶盖方腔驱动流问题上对该算法进行验证。实验结果表明,TSFEM较传统有限元法（FEM）无论是收敛速度还是计算效率都有了显著提高。     

两柔性机器人协调操作的动力学模型及其逆动力学分析

柔性机器人动力学是当前机器人研究的热点,而其协调操作问题目前仍为空白．本 文首次建立了柔性机器人协调操作刚性负载的动力学模型,利用有限元法和Lagrange方程, 在柔性机器人协调操作的运动学和动力学协调约束条件基础上,推导出系统的动力学方程, 提出了其逆动力学问题的解决方案,并成功给出了平面两3R柔性臂协调操作的数值算例．     

Studied X-FEM enrichment to handle material interfaces with higher order finite element

An approach to improve the geometrical representation of surfaces with the eXtended Finite Element Method is proposed. Surfaces are implicitly represented using the level set method. The finite element approximation is enriched by additional functions through the notion of partition of unity, to track material interfaces. Optimal rate of convergence is achieved with curved geometries, using linear elements and linear level set in elements. In order to accelerate the convergence, the order of approximation shape functions is increased, while keeping the same computational mesh. The level set is represented on a finer sub-mesh than the finite element mesh. A special attention to integration procedure is necessary. A new enrichment function is introduced to represent the behavior of curved material interfaces. Numerical examples including free surfaces and material interfaces in 2-D linear elasticity are presented to study convergence rates.     

A meshless method for numerical solution of the coupled Schrödinger-KdV equations

This paper presents meshless method using RBF collocation scheme for the coupled Schrödinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L ₂ and L _∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.     

A study on X-FEM in continuum structural optimization using a level set model

In this paper, we implement the extended finite element method (X-FEM) combined with the level set method to solve structural shape and topology optimization problems. Numerical comparisons with the conventional finite element method in a fixed grid show that the X-FEM leads to more accurate results without increasing the mesh density and the degrees of freedom. Furthermore, the mesh in X-FEM is independent of the physical boundary of the design, so there is no need for remeshing during the optimization process. Numerical examples of mean compliance minimization in 2D are studied in regard to efficiency, convergence and accuracy. The results suggest that combining the X-FEM for structural analysis with the level set based boundary representation is a promising approach for continuum structural optimization.     

The finite cell method for nearly incompressible finite strain plasticity problems with complex geometries

In this paper, the performance of the Finite Cell Method is studied for nearly incompressible finite strain plasticity problems. The Finite Cell Method is a combination of the fictitious domain approach with the high-order Finite Element Method. It provides easy mesh generation capabilities for highly complex geometries; moreover, this method offers high convergence rates, the possibility to overcome locking and robustness against high mesh distortions. The performance of this method is numerically investigated based on computations of benchmark and applied problems. The results are also verified with the $h$- and $p$-version Finite Element Method. It is demonstrated that the Finite Cell Method is an appropriate simulation tool for large plastic deformations of structures with complex geometries and microstructured materials, such as porous and cellular metals that are made up of ductile materials obeying nearly incompressible $J_2$ theory of plasticity.     

NURBS-Enhanced Finite Element Method (NEFEM)

The development of NURBS-Enhanced Finite Element Method (NEFEM) is revisited. This technique allows a seamless integration of the CAD boundary representation of the domain and the finite element method (FEM). The importance of the geometrical model in finite element simulations is addressed and the benefits and potential of NEFEM are discussed and compared with respect to other curved finite element techniques.     

最小二乘有限元法和有限体积法在CFD中的应用比较

为比较最小二乘有限元法（Least Square Finite Element Method,LSFEM）和有限体积法在CFD应用中的优劣,采用最小二乘法离散不可压N-S方程的有限元模型,得到正定对称线性系统,采用高效的预处理共轭梯度法求解方程组;利用LSFEM和基于有限体积法的FLUENT分别计算Kovasznay流动、定常二维和三维后台阶流动以及非定常圆柱绕流等4个实例并比较计算结果.结果表明,LSFEM比有限体积法的收敛性和精确性更好,在CFD领域的应用价值很高.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.