The finite element method (FEM) has very broad applications in a lot of research areas, and isogeometric analysis (IGA) is a new advancement based on FEM to integrate design with analysis. This chapter reviews the basic algorithm of finite element analysis (FEA) and its new developments, including IGA, extended FEM and immersed FEM. As a popular and powerful numerical method to solve partial differential equations over complex domains, FEM has been developed rapidly and used in many research areas including computational medicine, biology and engineering. The FEM is a general technique to solve boundary value problems with uniformly and non-uniformly spaced grids or meshes. In the implementation, the element stiffness matrix and element load vector are computed element by element, and then assembled together into the global stiffness matrix and global load vector. FEA has been applied …

This article describes the use of fluid flow simulation on parallel computing platforms to solve design problems in the automotive industry. The fluid flow formulation in the Spectrum TM solver, which is used for these simulations, is described. The finite element treatment of fluid flow in this work is based on the Galerkin-Least-Squares (GLS) method with discontinuity capturing operators. The Arbitrary-Lagrangian–Eulerian (ALE) method is utilized to account for deformable fluid domains. Automatically generated tetrahedral grids are used to ease and expedite the analysis process. The multiphysics architecture used in this work lends itself naturally to high-performance parallel computing. By taking advantage of automatic mesh generation and parallel computing, dramatic reduction in turnaround time for flow analysis is achieved. Several applications are presented which demonstrate the utility and accuracy of finite element solutions in automotive engineering problems and highlight the scalability of the software.

This work approaches strain localization by recognizing the multiple scales inherent in the problem. A component associated with the region of localized strain (typically, one with high gradient) is referred to as the fine scale. It represents the microstructure. The field obtained by removing the fine scale from the total solution is referred to as the coarse scale. The aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale. This process eliminates the fine scale from the problem, yet retains its effect. In applying this framework to the nonlinear problems with which localized strains are associated, a crucial step is a first-order approximation of the relevant relations. The fully nonlinear problem is solved by an iterative scheme. By accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization. It thus presents itself as a unifying framework for such models. Numerical solutions are shown to be invariant with respect to the discretization. Furthermore, for cases in which the displacements assume a distinct profile within the localization band, the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible. The model is applied to strain localization problems that arise in inviscid and viscoplastic solids. Numerical simulations are presented that demonstrate the efficacy of the approach.

In this paper we show the equivalence between the variational multiscale and the residual-free bubbles concepts.

In this paper we carry out adaptive finite element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate potentially significant cost savings engendered through adaptivity for propagating solutions at moderate wave numbers (ka = 2π, 4π). The computations are performed on meshes of linear triangles, and are adapted to the solution by locally changing element sizes (h-refinement). The adaptive procedure involves applying an explicit, residual-based a posteriori error estimator and an h-adaptive strategy, which were derived in a previous paper. The adaptive meshes are then obtained through global mesh regeneration using an advancing front mesh generator. Two different finite element formulations are considered: The Galerkin and Galerkin Least-Squares (GLS) methods. The infinite physical domain is truncated by an artificial exterior boundary, on which a fully coupled Dirichlet-to-Neumann (DtN) boundary condition is applied. Two different problems are computed: Plane wave scattering by a rigid infinite surface (circular and square cross sections), and nonuniform radiation from an infinite circular cylinder. Detailed cost studies with respect to an active column direct solver are performed. For the nonuniform radiation problem, the adaptive mesh is twenty times more cost effective than a uniform mesh (for Galerkin). When coupled with the GLS formulation, the adaptive mesh is forty times more efficient than Galerkin computations on a uniform mesh.

A space-time Galerkin/least-squares finite element formulation of the Navier-Stokes equations is presented for the analysis of free surface flows, moving spatial configurations and deforming fluid-structure interfaces. The variational equation is based on the time discontinuous Galerkin method employing the physical entropy variables. The space-time elements are oriented in time to accommodate the spatial deformations. If the elements are oriented along the particle paths, the formulation is Lagrangian and if they are fixed in time, it is Eulerian. Consequently this formulation is analogous to the arbitrary Lagrangian-Eulerian (ALE) technique. A novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration. Numerical results are presented to show the performance of the method.