A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem

Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG-dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.     

A coupled finite element-differential quadrature element method and its accuracy for moving load problem

In this paper a mixed method, which combines the finite element method and the differential quadrature element method (DQEM), is presented for solving the time dependent problems. In this study, the finite element method is first used to discretize the spatial domain. The DQEM is then employed as a step-by-step DQM in time domain to solve the resulting initial value problem. The resulting algebraic equations can be solved by either direct or iterative methods. Two general formulations using the DQM are also presented for solving a system of linear second-order ordinary differential equations in time. The application of the formulation is then shown by solving a sample moving load problem. Numerical results show that the present mixed method is very efficient and reliable.     

A Finite Element Approach for the Simulation of Quasi-Brittle Fracture

In the context of a strong discontinuity approach, we propose a finite element formulation with an embedded displacement discontinuity. The basic assumption of the proposed approach is the additive split of the total displacement field in a continuous and a discontinuous part. An arbitrary crack splits the linear triangular finite element into two parts, namely a triangular and a quadrilateral part. The discontinuous part of the displacement field in the quadrilateral portion is approximated using linear shape functions. For these purposes, the quadrilateral portion is divided into two triangular parts which is in this way similar to the approach proposed in [5]. In contrast, the discretisation is different compared to formulations proposed in [1] and [3], where the discontinuous part of the displacement field is approximated using bilinear shape functions. The basic theory of the underlying finite element formulation and a cohesive interface model to simulate brittle fracture are presented. By means of representative numerical examples differences and similarities of the present formulation and the formulations proposed in [1] and [3] are highlighted. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An object-oriented hybrid symbolic/numerical approach for the development of finite element codes

The association of object-oriented programming and symbolic computation techniques introduces certain changes in finite element code organization. The purpose of this approach is to speed up the design of new formulations. Previous papers have described the basic concepts of the method. In this paper, the focus is placed on functional aspects of symbolic tools for the development of finite element formulations. Two practical examples are used to illustrate this point. The first is a space-time formulation for an incompressible flow driven by the Navier–Stokes equations, and the second is a finite element derivation of the total potential energy for linear elasticity.     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

Finite elements using absolute nodal coordinates for large-deformation flexible multibody dynamics

A family of structural finite elements using a modern absolute nodal coordinate formulation (ANCF) is discussed in the paper with many applications. This approach has been initiated in 1996 by A. Shabana. It introduces large displacements of 2D/3D finite elements relative to the global reference frame without using any local frame. The elements employ finite slopes as nodal variables and can be considered as generalizations of ordinary finite elements that use infinitesimal slopes. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices and generalized gravity forces as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This approach allows applying known abstractions of real elastic bodies: Euler–Bernoulli beams, Timoshenko beams and more general models as well as Kirchhoff and Mindlin plate theories.     

Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.     

An enhanced finite volume method to model 2D linear elastic structures

This paper details the evaluation and enhancement of the vertex-centred finite volume method for the purpose of modelling linear elastic structures undergoing bending. A matrix-free edge-based finite volume procedure is discussed and compared with the traditional isoparametric finite element method via application to a number of test-cases. It is demonstrated that the standard finite volume approach exhibits similar disadvantages to the linear Q4 finite element formulation when modelling bending. An enhanced finite volume approach is proposed to circumvent this and a rigorous error analysis conducted. It is demonstrated that the developed finite volume method is superior to both standard finite volume and Q4 finite element methods, and provides a practical alternative to the analysis of bending-dominated solid mechanics problems.     

Wavelet-Galerkin method for solving parabolic equations in finite domains

A novel wavelet-Galerkin method tailored to solve parabolic equations in finite domains is presented. The emphasis of the paper is on the development of the discretization formulations that are specific to finite domain parabolic equations with arbitrary boundary conditions based on weak form functionals. The proposed method also deals with the development of algorithms for computing the associated connection coefficients at arbitrary points. Here the Lagrange multiplier method is used to enforce the essential boundary conditions. The numerical results on a two-dimensional transient heat conducting problem are used to validate the proposed wavelet-Galerkin algorithm as an effective numerical method to solve finite domain parabolic equations.     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．     

Transient elastodynamic three-dimensional problems in cracked bodies

The general formulation of the transient elastodynamic second boundary value problem in an isotropic linear elastic body with a crack of arbitrary shape by combining the boundary integral equation method and the Laplace transform with respect to time is presented in this paper. Both finite and infinite elastic bodies are considered. A numerical solution of the transformed boundary integral equations is proposed.     

Polynomial shape functions on the logarithmic space: the LogFE method

We extend the Logarithmic finite element method, a novel finite element approach for solving boundary-value problems proposed in [1], to a complete set of degrees of freedom, i.e. translational and rotational degrees of freedom in three dimensions. In contrast to the standard Ritz-Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function. Unlike existing formulations based on Lie groups, they may include polynomial functions of arbitrary degree. The method focuses on reducing the low-frequency components in the error, while minimizing spurious high-frequency deformations, a characteristic that is particularly advantageous in the context of a multigrid algorithm, in which the method may be used to construct an approximation for the coarse grid. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element method for contact/impact

Ideas from the analysis of differential-algebraic equations are applied to the numerical solution of frictionless contact/impact problems in solid mechanics. Index-one and two formulations for dynamic contact–impact within the context of the finite element method are derived. The resulting equations are shown to stabilize the kinematic fields at the contact interface, at the expense of a small energy loss, which is shown to decrease consistently with mesh refinement. This energy dissipation is shown to be necessary for the establishment of persistent contact. A Newmark-type time integration scheme is derived from the proposed formulation, and shown to yield excellent results in modeling the transition to contact/impact.     

An $L^{\infty}$ Second Order Cartesian Method for 3D Anisotropic Interface Problems

A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite element-finite difference (FE-FD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the point-wise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis.     

Nonanticipative duality,relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.     

Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.     

Comparison of light transport models based on finite element and the discrete ordinates methods in view of optical tomography applications

Numerical tests are used to evaluate the accuracy of two finite element formulations associated with the discrete ordinates method for solving the radiative transfer equation: the Least Square and the Discontinuous Galerkin finite element formulations. The results show that the use of a penalization method to set the Dirichlet boundary conditions leads to a more accurate solution than the weakly type setting where the Least Square method is seen to be more sensitive. Convergence in mesh size shows that, while both methods give accurate results, the Discontinuous Galerkin formulation uses five times more degrees of freedom than the Least Square formulation, which may lead to large systems to handle when the number of mesh elements is large. The comparison of both methods using the S_n and the T_n angular quadratures has shown that the Discontinuous Galerkin gives more accurate solutions, as expected, for problems with strong discontinuities, but may exhibit some oscillations due to the Galerkin procedure. A last test featuring a collimated irradiation shows that both methods give the same accuracy due to the separation of the radiative intensity into transmitted and scattered components, which removes the discontinuities in the implementation of the boundary conditions.     

A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full 3D finite element Cosserat formulation with application in layered structures

In this paper, a full three-dimensional (3D) finite element Cosserat formulation is developed within the principles of continuum mechanics in the small deformation framework. The developed finite element formulation is general; however, the proposed constitutive laws incorporate the effect of the internal length parameter of 3D layered continua. The extension of the existing two-dimensional (2D) Cosserat formulation to the 3D framework is novel and is consistent with plate theory which can be considered as the 3D version of beam theory. The results demonstrate a high level of consistency with the analytical solutions predicted by plate theory as well as predictions by alternative numerical techniques such as the discrete element method.