Imposition of essential boundary conditions in the material point method

There is increasing interest in the material point method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming; however, some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper, we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, for the MPM, expanding its capabilities to model boundaries of any inclination. To the authors' knowledge, this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and nonzero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that (1) the boundaries are independent of the background mesh, (2) artificially stiff regions of material points are avoided, and (3) the method does not rely on mirroring of the problem domain to impose symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM.     

Trefftz boundary elements—multi‐region formulations

This paper is concerned with an effective numerical implementation of the Trefftz boundary element method, for the analysis of two‐dimensional potential problems, defined in arbitrarily shaped domains. The domain is first discretized into multiple subdomains or regions. Each region is treated as a single domain, either finite or infinite, for which a complete set of solutions of the problem is known in the form of an expansion with unknown coefficients. Through the use of weighted residuals, this solution expansion is then forced to satisfy the boundary conditions of the actual domain of the problem, leading thus to a system of equations, from which the unknowns can be readily determined. When this basic procedure is adopted, in the analysis of multiple‐region problems, proper boundary integral equations must be used, along common region interfaces, in order to couple to each other the unknowns of the solution expansions relative to the neighbouring regions. These boundary integrals are obtained from weighted residuals of the coupling conditions which allow the implementation of any order of continuity of the potential field, across the interface boundary, between neighbouring regions. The technique used in the formulation of the region‐coupling conditions drives the performance of the Trefftz boundary element method. While both of the collocation and Galerkin techniques do not generate new unknowns in the problem, the technique of Galerkin presents an additional and unique feature: the size of the matrix of the final algebraic system of equations which is always square and symmetric, does not depend on the number of boundary elements used in the discretization of both the actual and region‐interface boundaries. This feature which is not shared by other numerical methods, allows the Galerkin technique of the Trefftz boundary element method to be effectively applied to problems with multiple regions, as a simple, economic and accurate solution technique. A very difficult example is analysed with this procedure. The accuracy and efficiency of the implementations described herein make the Trefftz boundary element method ideal for the study of potential problems in general arbitrarily‐shaped two‐dimensional domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite cell method

A simple yet effective modification to the standard finite element method is presented in this paper. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. If this extension is smooth, the extended solution can be well approximated by high order polynomials. This way, the finite element mesh can be replaced by structured or unstructured cells embedding the domain where classical h- or p-Ansatz functions are defined. An adequate scheme for numerical integration has to be used to differentiate between inside and outside the physical domain, very similar to strategies used in the level set method. In contrast to earlier works, e.g., the extended or the generalized finite element method, no special interpolation function is introduced for enrichment purposes. Nevertheless, when using p-extension, the method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities. The formulation in this paper is applied to linear elasticity problems and examined for 2D cases, although the concepts are generally valid. The first author would like to appreciate the financial support of his stay in Germany, where this research has been carried out, by the Alexander von Humboldt foundation.     

Multiple holes,cracks, and inclusions in anisotropic viscoelastic solids

By using the elastic–viscoelastic correspondence principle, the problems with multiple holes, cracks, and inclusions in two-dimensional anisotropic viscoelastic solids are solved for the cases with time-invariant boundaries. Based upon this principle and the existing methods for the problems with anisotropic elastic materials, two different approaches are proposed in this paper. One is concerned with an analytical solution for certain specific cases such as two collinear cracks, collinear periodic cracks, and interaction between inclusion and crack, and the other is a boundary-based finite element method for the general cases with multiple holes, cracks, and inclusions. The former considers only specific cases in infinite domain and can be used as a reference for any other numerical methods, and the latter is applicable to any combination of holes, cracks and inclusions in finite domain, whose number, size and orientation are not restricted. Unlike the conventional finite element method or boundary element method which usually needs very fine meshes to get convergence solutions, in the proposed boundary-based finite element method no meshes are needed along the boundaries of holes, cracks and inclusions. To show the accuracy and efficiency of these two proposed approaches, several representative examples are implemented analytically and numerically, and they are compared with each other or with the solutions obtained by the finite element method.     

The w-plane finite element method for the solution of scalar field problems in two dimensions

A simple, efficient finite element method has been presented for the solution of a variety of scalar field problems in two dimensions. It is based on the mapping of the physical problem domain into an ‘image’ domain in the w-plane. The governing equation(s) and the boundary conditions in the physical plane are also appropriately transformed into the w-plane. The processes of standard finite element analysis are then implemented to obtain a solution in the w-plane. The method has been explained in detail, with illustrative examples where appropriate; it has several important advantages over the standard finite element method, particularly for the solution of infinite or semi-infinite domain problems. The method has been demonstrated to be simple, efficient, economical and potentially capable of dealing with a large repartoire of two-dimensional problems, including non-homogeneity, nonlinearity, etc.     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

A Burton–Miller formulation of the boundary element method for baffle problems in acoustics and the BEM/FEM coupling

Baffle problems, i.e. radiation problems from objects mounted behind a hole of an infinite hard reflecting wall, can be simulated as a multi domain problem consisting of a finite interior domain around the object, and two infinite half spaces in front and behind the baffle plane. A formulation of such problems is presented in the context of the Burton–Miller boundary element method. Additionally, the coupling of the acoustic boundary element method and the structural finite element method in the context of the Burton–Miller-formulation of the baffle problem is discussed.     

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian-grid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary-fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q₁-non-conforming finite element method is analyzed for second-order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q₁-rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non-conforming. A stair-step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non-boundary-fitted stair-stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in 𝒪(h^1/2) for the H¹-norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L²-norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non-boundary-fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

Solving boundary-value problems using hp-version finite elements in time

An hp-version finite element method for one-dimensional boundary value problems is presented. The method is based on a similar approach developed by the authors for solution of optimal control problems. The primary applications for the methodology include two-point- and multi-point-boundary-value problems, for example, in the time domain. Results presented for a 7-state/3-phase missile problem show that the method is very efficient for time-marching applications. Furthermore, it easily solves time-domain problems with discontinuities in the system equations and/or in the states, where the time at which these jumps (i.e. ‘events’) take place is determined by equations that govern the states. An example involving friction with intermittent sticking is presented to illustrate the power of the method. © 1998 John Wiley & Sons, Ltd.     

Coons-patch macroelements in potential Robin problems

This paper investigates the performance of large isoparametric finite elements based on the Coons' patch interpolation formula in the analysis of two-dimensional and axisymmetric potential problems with mixed (Robin) boundary conditions. This formula allows the global interpolation of the potential, e.g. temperature, within the whole domain and leads to the so-called “macroelements”, where the degrees of freedom appear only at the element boundaries. Numerical results including a steady-state axisymmetric thermal problem and simple test problems in two dimensions from literature, sustain the proposed method, which is successfully compared with conventional finite elements, boundary elements and exact analytical solutions.     

Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization

We consider the numerical simulation of non‐linear multi‐body contact problems in elasticity on complex three‐dimensional geometries. In the case of warped contact boundaries and non‐matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non‐penetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non‐conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element‐based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non‐linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach. Copyright © 2008 John Wiley & Sons, Ltd.     

A hybrid finite difference and moving least square method for elasticity problems

In this paper, a novel hybrid finite difference and moving least square (MLS) technique is presented for the two-dimensional elasticity problems. A new approach for an indirect evaluation of second order and higher order derivatives of the MLS shape functions at field points is developed. As derivatives are obtained from a local approximation, the proposed method is computationally economical and efficient. The classical central finite difference formulas are used at domain collocation points with finite difference grids for regular boundaries and boundary conditions are represented using a moving least square approximation. For irregular shape problems, a point collocation method (PCM) is applied at points that are close to irregular boundaries. Neither the connectivity of mesh in the domain/boundary or integrations with fundamental/particular solutions is required in this approach. The application of the hybrid method to two-dimensional elastostatic and elastodynamic problems is presented and comparisons are made with the boundary element method and analytical solutions.     

Enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions

Patch recovery based on superconvergent derivatives and equilibrium (SPRE), an enhancement of the Superconvergent Patch Recovery (SPR), is studied for linear elasticity problems. The paper also presents a further improvement for recovery of derivatives near boundaries, SPREB, where either tractions or displacements are prescribed. This is made by inclusion of weighted residual errors at boundary points in the patch recovery. A pronounced improvement in the post processed gradients of the finite element solution is observed by this method.     

ON OPEN BOUNDARIES IN THE FINITE ELEMENT APPROXIMATION OF TWO-DIMENSIONAL ADVECTION–DIFFUSION FLOWS

A steady-state and transient finite element model has been developed to approximate, with simple triangular elements, the two-dimensional advection–diffusion equation for practical river surface flow simulations. Essentially, the space–time Crank–Nicolson–Galerkin formulation scheme was used to solve for a given conservative flow-field. Several kinds of point sources and boundary conditions, namely Cauchy and Open, were theoretically and numerically analysed. Steady-state and transient numerical tests investigated the accuracy of boundary conditions on inflow, noflow and outflow boundaries where diffusion is important (diffusive boundaries). With the proper choice of boundary conditions, the steady-state Galerkin and the transient Crank–Nicolson–Galerkin finite element schemes gave stable and precise results for advection-dominated transport problems. Comparisons indicated that the present approach can give equivalent or more precise results than other streamline upwind and high-order time-stepping schemes. Diffusive boundaries can be treated with Cauchy conditions when the flow enters the domain (inflow), and with Open conditions when the flow leaves the domain (outflow), or when it is parallel to the boundary (noflow). Although systems with mainly diffusive noflow boundaries may still be solved precisely with Open conditions, they are more susceptible to be influenced by other numerical sources of error. Moreover, the treatment of open boundaries greatly increases the possibilities of correctly modelling restricted domains of actual and numerical interest. © 1997 by John Wiley & Sons, Ltd.     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

BOUNDARY ELEMENT–FINITE ELEMENT COUPLED EIGENANALYSIS OF FLUID–STRUCTURE SYSTEMS

The eigenanalysis of acoustical cavities with flexible structure boundaries, such as a fluid-filled container or an automobile cabin enclosure, is considered. An algebraic eigenvalue problem formulation for the fluid–structure problem is presented by combining the acoustic fluid boundary element eigenvalue analysis method and the structural finite elements. For many practical eigenproblems, use of finite elements to discretize the fluid domain leads to large stiffness and mass matrices. Since the acoustic boundary element discretization requires putting nodes only on the wetted surface of the structure, the size of the eigenproblem is reduced considerably, thus reducing the eigenvalue extraction effort. Futhermore, unlike in ordinary cases, the finite element discretization of pressure–displacement based fluid–structure problem gives rise to unsymmetric matrices. Therefore, the fact that the boundary element formulation produces unsymmetric matrices does not introduce additional difficulties here compared to the finite element case in the choice of an eigenvalue extraction procedure. Examples are included to demonstrate the fluid–structure eigenanalysis using boundary elements for the fluid domain and finite elements for the structure.     

An effective hybrid displacement function element method for solving the edge effect of Mindlin–Reissner plate

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 β [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. Copyright © 2015 John Wiley & Sons, Ltd.     

Continuously deforming finite elements for the solution of parabolic problems,with and without phase change

A number of transport problems are complicated by the presence of physically important transition zones where quantities exhibit steep gradients and special numerical care is required. When the location of such a transition zone changes as the solution evolves through time, use of a deforming numerical mesh is appropriate in order to preserve the proper numerical features both within the transition zone and at its boundaries. A general finite element solution method is described wherein the elements are allowed to deform continuously, and the effects of this deformation are accounted for exactly. The method is based on the Galerkin approximation in space, and uses finite difference approximations for the time derivatives. In the absence of element deformation, the method reduces to the conventional Galerkin formulation. The method is applied to the two-phase Stefan problem associated with the melting and solidification of A substance. The interface between the solid and liquid phase form an internal moving boundary, and latent heat effects are accounted for in the associated boundary condition. By allowing continuous mesh deformation, as dictated by this boundary condition, the moving boundary always lies on element boundaries. This circumvents the difficulties inherent in interpolation of parameters and dependent variables across regions where those quantities change abruptly. Basis functions based on Hermite polynomials are used, to allow exact specification of the flux-latent heat balance condition at the phase boundary. Analytic solutions for special cases provide tests of the method.     

Generalization of robustness test procedure for error estimators. Part II: test results for error estimators using SPR and REP

In this part of the paper we shall use the formulation given in the first part to assess the quality of recovery‐based error estimators using two recovery methods, i.e. superconvergent patch recovery (SPR) and recovery by equilibrium in patches (REP). The recovery methods have been shown to be asymptotically robust and superconvergent when applied to two‐dimensional problems. In this study we shall examine the behaviour of the recovery methods on several three‐dimensional mesh patterns for patches located either inside or at boundaries. This is performed by first finding an asymptotic finite element solution, irrespective of boundary conditions at far ends of the domain, and then applying the recovery methods. The test procedure near kinked boundaries is explained in a step‐by‐step manner. The results are given in a series of tables and figures for various cases of three‐dimensional mesh patterns. It has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range. Copyright © 2005 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.