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A FEM-DtN formulation for a non-linear exterior problem in incompressible elasticity
Abstract
In this paper, we combine the usual finite element method with a Dirichlet-to-Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non-linear incompressible 2d-elasticity. We show that the variational formulation can be written in a Stokes-type mixed form with a linear constraint and a non-linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea-type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.
... This article presents a procedure for studying the Galerkin approximation to an incompressible material on unbounded exterior domain Ω. This procedure used Dirichlet-to-Neumann mapping (DtN), (Han and Bao, 1997;Gatica, Gatica and Stephan, 2003;Kako and Touda, 2004), consisting of introducing an artificial boundary by drawing circumference Γ in ℝ with radius in domain Ω; Ω was then divided in bounded part Ω and unbounded part Ω . Exact and approximate boundary conditions were given on artificial boundary Γ to solve the problem on bounded domain ( Figure 1). ...
En este trabajo presentamos la implementación de un método de elementos finitos combinado con la aplicación Dirichlet-to-Neumann (D†N), obtenida en términos de series de Fourier, para estudiar la existencia de soluciones de un problema exterior que proviene de la teoría de elasticidad lineal incompresible bidimensional. Finalmente, se presenta un método numérico que demuestra la precisión de nuestra aplicación D†N, puesto que sólo se necesitan unos cuantos términos de la serie de Fourier para obtener una buena aproximación de la solución. Para la discretización del problema se emplea el elemento estable Taylor-Hood.
For the numerical treatment of elliptic boundary value problems, the coupling of boundary element methods (BEM) and finite element methods (FEM) is widely used. Usually, the FEM is applied in a bounded subdomain, whereas the BEM subdomain may be unbounded, but without inhomogeneous or nonlinear material behavior. This article gives a survey of FE/BE coupling for elliptic interface problems from a mathematical perspective. We concentrate on the symmetric coupling method because of its general applicability and give a priori and a posteriori error estimates. Further, we address implementational aspects such as suitable preconditioning for appropriate solvers of the Galerkin equations. As new developments, we consider a least squares coupling method with first-order systems in the FEM domain and consider both primal and dual FE/BE coupling methods for interface problems with Signorini contact condition. Besides standard formulations, mixed FEM formulations are also considered. Applications in elasticity are discussed. We present proofs of essential theorems because we feel that these proofs are essential for understanding the mathematical aspects. We also present illustrative numerical examples and provide a list of references to the current literature. The new version contains sections on convergence of an adaptive coupling algorithm and on nonsymmetric coupling methods
The Schwarz alternating algorithm, which is based on natural boundary element method, is constructed for solving the exterior anisotropic problem in the three-dimension domain. The anisotropic problem is transformed into harmonic problem by using the coordinate transformation. Correspondingly, the algorithm is also changed. Continually, we analysis the convergence and the error estimate of the algorithm. Meanwhile, we give the contraction factor for the convergence. Finally, some numerical examples are computed to show the efficiency of this algorithm.
In this chapter, we discuss the implicit ABCs for the exterior problem of 2-D and 3-D Poisson equations, the Helmholtz equation, and the Navier system, and for the wave equation on unbounded domains. By using artificial boundaries, the original problems are reduced to boundary or initial boundary value problems on bounded computational domains. Implicit boundary conditions on the artificial boundaries are obtained, and then the finite element or finite difference method is applied to solve the reduced problems. Some error estimates are also given.
In this chapter, we discuss the nonlinear ABCs for Burgers equation, Kardar-Parisi-Zhang equation, and Schrödinger equation on unbounded domains. By using artificial boundaries, the original problems are reduced to initial boundary value problems on bounded computational domains. Nonlinear boundary conditions on the artificial boundaries are obtained, and then the finite difference method is applied to solve the reduced problems. Some stability results are also given.
An artificial boundary condition method, derived in terms of infinite Fourier series, is applied to solve a class of quasi-Newtonian Stokes flows. Based on the natural boundary reduction involving an artificial condition on the artificial boundary, the coupled variational problem and its numerical solution are obtained. The unique solvability of the continuous and discrete formulations are discussed, and the error analysis for the problem is also considered. Finally, an a posteriori error estimate for the corresponding problem is provided.
"Artificial Boundary Method" systematically introduces the artificial boundary method for the numerical solutions of partial differential equations in unbounded domains. Detailed discussions treat different types of problems, including Laplace, Helmholtz, heat, Schrödinger, and Navier and Stokes equations. Both numerical methods and error analysis are discussed. The book is intended for researchers working in the fields of computational mathematics and mechanical engineering. © 2013 Tsinghua University Press, Beijing and Springer-Verlag Berlin Heidelberg. All rights are reserved.
The artificial boundary method is applied to solve three-dimensional exterior problems. Two kinds of rotating ellipsoids are chosen as the artificial boundaries and the exact artificial boundary conditions are derived explicitly in terms of an infinite series. Then, the well-posedness of the coupled variational problem is obtained. It is found that the error estimates derived depend on the mesh size, the truncation term and the location of the artificial boundary. Three numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.
The artificial boundary method is one of the most popular and effective numerical methods for solving partial differential equations on unbounded domains, with more than thirty years development. The artificial boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.
In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.
This paper deals with a least-squares formulation of a second order transmission problem for linear elasticity. The problem in the unbounded exterior domain is rewritten with boundary integral equations on the boundary of the inner domain. In the interior domain we treat a linear elastic material which can also be nearly incompressible. The least-squares functional is given in terms of the H˜−1(Ω) and H1/2(Γ)H1/2(Γ) norms. These norms are realized by solution operators of corresponding dual norm problems which are approximated using multilevel preconditioners.
In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dual–dual ones), which are analyzed by means of a slight generalization of the well known Babuška–Brezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, Raviart–Thomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces.
For the numerical treatment of elliptic boundary value problems, the coupling of boundary element methods and finite element methods is widely used. Usually, the FEM is applied in a bounded subdomain, whereas the BEM subdomain may be unbounded, but without inhomogeneous or nonlinear material behavior. This chapter gives a survey of FE/BE coupling for elliptic interface problems with emphasis on mathematics. We concentrate on the symmetric coupling method because of its general applicability and give a priori and a posteriori error estimates. Further, we address implementational aspects such as suitable preconditioning for appropriate solvers of the Galerkin equations. As new developments, we consider a least squares coupling method with first-order systems in the FEM domain and consider both primal and dual FE/BE coupling methods for interface problems with Signorini contact condition. Besides standard formulations, mixed FEM formulations are also considered. Applications in elasticity are discussed. We present proofs of essential theorems because we feel that these proofs are essential for understanding the mathematical aspects. We also present illustrative numerical examples and provide a list of references to the current literature.
Keywords:
boundary elements;
finite elements;
coupling;
a posteriori error estimates;
fast solvers;
least squares methods;
Signorini problems;
nonlinear elasticity;
primal- and dual-mixed coupling methods
Introduction Electroless deposition: basic processMixed potential theory and electroless deposition General principlesMixed potential theory: experimental observationsCatalytic and mechanistic aspects of electroless deposition Catalytic aspectsMechanistic overview Hydrogen evolution in electroless depositionElectroless Ni and CoElectroless Cu Alternative reducing agents to formaldehydeAlloy deposition Alloys not containing refractory metalsAlloys containing catalytically inactive metalsSolution composition: effect of solution stability, deposition kinetics, and deposit uniformity Metal ion ComplexationConsequences of metal ion diffusion controlSolution additivesDissolved oxygen gasConclusions Applied considerationsFundamental considerations Electroless deposition: basic process General principlesMixed potential theory: experimental observations Catalytic aspectsMechanistic overview Hydrogen evolution in electroless depositionElectroless Ni and CoElectroless Cu Alternative reducing agents to formaldehydeAlloy deposition Alloys not containing refractory metalsAlloys containing catalytically inactive metals Hydrogen evolution in electroless deposition Alternative reducing agents to formaldehyde Alloys not containing refractory metalsAlloys containing catalytically inactive metals Metal ion ComplexationConsequences of metal ion diffusion controlSolution additivesDissolved oxygen gas ComplexationConsequences of metal ion diffusion control Applied considerationsFundamental considerations
We analyze a least squares formulation for the numerical solution of second-order lineartransmission problems in two and three dimensions, which allow jumps on the interface. In a bounded domain the second-order partial differential equation is rewritten as a first-order system; the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface. The least squares functional is given in terms of Sobolev norms of order -1 and of order 1/2. These norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second-order elliptic problem in a bounded domain Ω and for the weakly singular integral operator of the single layer potential on its boundary ∂Ω. As preconditioners we use both multigrid and BPX algorithms, and the preconditioned system has bounded or mildly growing condition number. Numerical experiments confirm our theoretical results.
A Neumann boundary value problem of plane elasticity problem in the exterior circular domain is reduced into an equivalent natural boundary integral equation and a Poisson integral formula with the DtN method. Using the trigonometric wavelets and Galerkin method, we obtain a fast numerical method for the natural boundary integral equation which has an unique solution in the quotient space. We decompose the stiffness matrix in our numerical method into four circulant and symmetrical or antisymmetrical submatrices, and hence the solution of the associated linear algebraic system can be solved with the fast Fourier transform (FFT) and the inverse fast Fourier transform (IFFT) instead of the inverse matrix. Examples are given for demonstrating our method has good accuracy of our method even though the exact solution is almost singular.
We combine a dual-mixed finite element method with the boundary integral equation method, to study the solvability and Galerkin approximations of a linear-nonlinear transmission problem arising in plane elastostatics. Our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain, and it leads to what we call a dual-dual mixed formulation since the corresponding operator equation becomes of a twofold saddle-point structure. We derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis. In particular, we extend the finite element space for plane elasticity given by the PEERS element and define an explicit finite element/boundary element subspace satisfying all the required compatibility conditions. Most of our analysis makes use of an extension of the classical Babuska--Brezzi theory to a class of nonlinear saddle-point problems.
A class of transmission problems is considered in which a nonlinear variational problem in one domain is coupled with a linear elliptic problem in a second domain. A typical example is a problem from three-dimensional elasticity theory where an elastoplastic material is embedded into a linear elastic material. The nonlinear problem is given in variational form with a strictly convex functional. The linear elliptic problem is described by boundary integral equations on the coupling boundary. The typical saddle point structure of such problems is analyzed. Galerkin approximations are studied which consist of a finite element approximation in the first domain coupled with a boundary element method on the coupling boundary. The convergence of the Galerkin approximation is based on the saddle-point structure which is shown to hold for the exact as well as the discretized problems.
This paper presents a new method for solving a nonlinear exterior boundary value problem arising in two-dimensional elasto-plasticity. The procedure is based on the introduction of a sufficiently large circle that divides the exterior domain into a bounded region and an unbounded one. This allows us to consider the Dirichlet-Neumann mapping on the circle, which provides an explicit formula for the stress in terms of the displacement by using an appropriate infinite Fourier series. In this way we can reduce the original problem to an equivalent nonlinear boundary value problem on the bounded domain with a natural boundary condition on the circle. Hence, the resulting weak formulation includes boundary and field terms, which yields the so called boundary-field equation method. Next, we employ the finite Fourier series to obtain a sequence of approximating nonlinear problems from which the actual Galerkin schemes are derived. Finally, we apply some tools from monotone operators to prove existence, uniqueness and approximation results, including Cea type error estimates for the corresponding discrete solutions.
We present a numerical method for solving an exterior Dirichlet problem in the plane. The technique consists in coupling boundary integral and mixed finite element methods. An artificial boundary is introduced separating an interior region from an exterior one. From an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary. These integral equations are incorporated into the so-called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem.
The aim of this paper is to give an elementary proof of a theorem of approximation of Sobolev spaces H**q( OMEGA ) by finite elements without using classical interpolation. The construction given makes it possible, in some cases, to fit boundary conditions.
The exterior boundary value problems of linear elastic equations are
considered. A sequence of approximations to the exact boundary
conditions at an artificial boundary is given. Then the original problem
is reduced to a boundary value problem on a bounded domain. Furthermore,
a finite element approximation of this problem and optimal error
estimates are obtained. Finally, a numerical example shows the
effectiveness of this method.
This paper considers the symmetric coupling method with boundary elements and finite elements analyzed by Costabel and Stephan. We present experimental rates of convergence for the h version, where we achieve the accuracy of the Galerkin approximation by refining the mesh. In our computations, we use as finite elements piecewise bilinear functions and as boundary elements piecewise constant functions.
The exterior boundary value problems of linear elastic equations are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem on a bounded domain. Furthermore, a finite element approximation of this problem and optimal error estimates are obtained. Finally, a numerical example shows the effectiveness of this method.
We recently have applied the coupling of boundary integral and finite element methods to study the weak solvability of certain nonlinear exterior boundary value problems. The main goal of this paper is to use an analogue of the coupling method to solve a three-dimensional elasto- plastic interface problem. For this purpose, we introduce and analyze a general class of variational formulations which allows us to extend the coupling technique to a wider class of nonlinear boundary value problems, including the elasto-plastic problem as a special example. Existence, uniqueness and approximation results for the weak formulation of this nonlinear interface problem are then deduced as a direct application of our generalized approach.
We apply the coupling of mixed finite elements and boundary integral methods, using two integral equations on the coupling boundary, to study the weak solvability of a nonlinear elliptic problem arising in elastostatics. Our procedure is based on both the usual Brezzi's theory for linear constrained variational problems and some fundamental tools from nonlinear functional analysis. We derive existence and uniqueness of solution for the continuous variational formulations and provide general approximation results for the corresponding Galerkin schemes. Although we consider bounded domains, the same analysis applies for the case in which the boundary element region is unbounded.
This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.
The mathematical basis of FEMs for incompressible steady interior flow
problems is examined in a text (a revised and expanded version of the
work of Girault and Raviart, 1979) intended for a postgraduate course in
numerical analysis. Chapters are devoted to the mathematical foundation
of the Stokes problem, results obtained with a standard FEM
approximation, the numerical solution of the Stokes problem in primitive
variables, incompressible mixed FEMs for solving the Stokes problem, and
the theory and approximation of the Navier-Stokes problem.
Ein gekoppeltes BEM/FEM-Problem aus der Elastostatik, ein typisches Substrukturproblem, wird im Rahmen der symmetrisch-definiten BEM behandelt. Es werden vier verschiedene Variationsformulierungen vorgestellt, in deren jeder die Transmissionsbedingungen gegenüber der Trennfläche zwischen FE-Unterregion und BE-Unterregion die Rolle natürlicher Randbedingungen spielen. Zwei der oben erwähnten Formulierungen sind Stationaritätsprinzipien in gemischter Form, die anderen beiden sind Sattelpunkt-Prinzipien, d. h. Kombinationen des Rand-min-max-Prinzips entweder mit dem Prinzip der minimalen Gesamtpotentialenergie oder mit dem Prinzip der minimalen Gesamtkomplementaritätsenergie. Jedes der oben angeführten gekoppelten Prinzipien kann für das durch unabhängige Modellierung für die Unbekannten auf der Trennfläche charakterisierte Diskretisierungsverfahren angewandt werden.
A coupled BEM/FEM problem in elastostatics, a typical substructuring problem, is addressed within the framework of the symmetric-definite BEM. Four different variational formulations are presented, in all of which the transmission conditions across the interface between the FE-subdomain and the BE-subdomain play the role of natural boundary conditions. Two of the above formulations are stationarity principles of mixed form, the other two are saddle-point principles, combinations of the boundary min-max principle with either the minimum total potential energy principle, or the minimum total complementary energy principle. Anyone of the above coupled principles can be employed for the discretization procedure characterized by independent modelling for the unknowns upon the interface.
The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes-type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi-optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001
We study the solvability and Galerkin approximation of an exterior hyperelastic interface problem arising in plane elasticity. The weak formulation is obtained from an appropriate combination of a mixed finite element approach with a Dirichlet-to-Neumann method. The derivation of our results is based on some tools from nonlinear functional analysis and the Babuska-Brezzi theory for variational problems with constraints.
Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to
a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in
elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed
in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis
of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with
a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical
examples are included.
In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an
unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically
by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective.
Incompressibility is gradually introduced into the finite elements with the mesh refinement in such a way as to balance it with the residual discretization energy in order to ensure fastest convergence to the incompressible solution with best conditioned stiffness matrix to minimize round-off errors in the computations.
The Finite Element Method
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Zienkiewicz OC. The Finite Element Method. MacGraw-Hill: London, 1977.
On simpliÿed hybrid methods for coupling of ÿnite elements and boundary elements
- Jj Grannell
Grannell JJ. On simpliÿed hybrid methods for coupling of ÿnite elements and boundary elements. In Boundary Elements IX, vol. 1, Brebbia CA et al. (eds). Springer: Berlin, 1987; 447 – 460.
Equation Methods for a Class of Nonlinear Problems
- Boundary-Field
- Gatica