Parallel Iterative Methods for the Helmholtz Equation With Exact Nonreflecting Boundaries

On the implementation of the Dirichlet-to-Neumann radiation condition for iterative solution of the Helmholtz equation

This communication describes an efficient implementation of the non-local Dirichlet-to-Neumann (DtN) radiation boundary condition which arises in the solution of exterior problems in acoustics. Exterior problems in acoustics involve unbounded fluid domains whose finite element solution requires the introduction of a truncation boundary in order to obtain a finite computational domain. The non-local (DtN) condition is an exact non-reflecting boundary condition which is imposed on this truncation boundary. Unfortunately, the discretization of the non-local (DtN) boundary condition results in a dense, fully populated matrix whose storage and factorization become increasingly expensive. We describe here a matrix-free interpretation of the non-local (DtN) map suitable for iterative solution methods, which allows the use of this exact boundary condition without any storage penalties related to its non-local nature.

The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.

State‐of‐the‐art computational methods for linear acoustics are reviewed. The equations of linear acoustics are summarized and then transformed to the frequency domain for time‐harmonic waves governed by the Helmholtz equation. Three major current challenges in the field are specifically addressed: numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales and requiring a large computational effort; the effective treatment of unbounded domains by domain‐based methods; and parallel iterative methods for large complex, ill‐conditioned and possibly indefinite equation systems for high‐frequency problems. A priori error estimates, including both dispersion (phase error) and global pollution effects for moderate to large wave numbers are discussed. Stabilized and other wave‐based discretization methods are reviewed. Domain‐based methods for modeling exterior domains are described including Dirichlet‐to‐Neumann (DtN) methods, absorbing boundary conditions, infinite elements, and the perfectly matched layer (PML). Efficient equation‐solving methods for the resulting complex‐symmetric (non‐Hermitian) matrix systems are discussed including parallel iterative methods and domain decomposition methods. Numerical methods for direct solution of the acoustic wave equation in the time domain are also reviewed.

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

A numerical implementation of the Domain Decomposition (DD) method for parallel computation of time-harmonic waves is presented. The Finite Element method on unstructured mesh is employed to discretize the boundary-value problem. A nonover-lapping mesh partitioning is used to build an overlapping domain decomposition. The algorithm also defines the new subdomain boundaries, other than the original boundaries of the global computational domain, on which transmission conditions are introduced. The independent subdomain problems can be solved concurrently, one problem per processor, on a parallel computer. The iterative scheme used to update the subdomain transmission conditions is computationally efficient and only swaps the nodal values in the overlapping region between adjacent subdomains. For exterior problems, the communication required to update the conditions on the outer boundary is substantially reduced by employing a modified Dirichlet-to-Neumann (DtN) map with truncated support. Numerical results are presented for a radiation model problem. It is shown that the DtN nonreflecting conditions eliminate spurious reflections and that the iterative scheme leads to a continuous global solution. The DD algorithm also has a very good convergence rate and significantly reduces memory requirement in comparison with a direct solver.

In this paper, a method based on the Dirichletto-Neumann map is developed for bandgap calculation of mixed in-plane waves propagating in 2D phononic crystals with square and triangular lattices. The method expresses the scattered fields in a unit cell as the cylindrical wave expansions and imposes the Bloch condition on the boundary of the unit cell. The Dirichlet-to-Neumann (DtN) map is applied to obtain a linear eigenvalue equation, from which the Bloch wave vectors along the irreducible Brillouin zone are calculated for a given frequency. Compared with other methods, the present method is memory-saving and time-saving. It can yield accurate results with fast convergence for various material combinations including those with large acoustic mismatch without extra computational cost. The method is also efficient for mixed fluid-solid systems because it considers the different wave modes in the fluid and solid as well as the proper fluid-solid interface condition.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

Finite element approximation of an elliptic variational inequality with quaisilinear operator and constraints to the solution and its gradient is constructed. Corresponding discrete problem is splitted into subproblems by non-overlapping domain decomposition technique and constrained saddle point problem is constructed for the splitted problem. Block relaxation-Uzawa iterative solution method is applied to this resulting saddle point problem. Existence of a solution to saddle point problem and convergence of the iterative method are proved.

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

The solution of exterior interface problems using a variational method with Lagrange multipliers

9R1. Natural Boundary Integral Method and Its Applications. - De-hao Yu (Inst of Comput Math and Sci/Eng Comput, Chinese Acad of Sci, Beijing, ROC). Kluwer Acad Publ, Dordrecht, Netherlands. Distributed in USA by Kluwer Acad Publ, Norwell MA. 2002. 539 pp. ISBN 1-4020-0457-5. $155.00.Reviewed by D Givoli (Dept of Aerospace Eng, Technion-Israel, Haifa, 32000, Israel).Kang Feng (1923-1993) was a well-known Chinese mathematician, and until his death was the President of the Chinese Society of Computational Mathematics. De-hao Yu, the author of this book, was his PhD student. This is an English version of a previous monograph by Yu that appeared in Chinese ten years ago. The book summarizes Yu’s research work based on the mathematical foundation laid by Feng. Most of Feng’s and Yu’s publications have appeared in Chinese and have not been generally accessible to western readers. This monograph, published by Kluwer, is especially welcome since it allows general access to the interesting work of both Feng and Yu. The book focuses on the concept of Natural Boundary Reduction (NBR) in the context of two-dimensional (2D) elliptic problems, and in particular problems governed by Laplace’s equation, the biharmonic equation, plane elasticity equations, and Stokes’ equations. The idea is to replace the differential equation in the given 2D domain Ω by an integral equation on its boundary Γ, and to solve this integral equation numerically on Γ via a variational formulation. This sounds like the basic idea of the Boundary Element Method (BEM); however, in NBR the reduction of the differential equation in Ω to an integral equation on Γ is performed differently, using the so-called natural integral operator, which is also known as the Dirichlet-to-Neumann (DtN) map. The book is written beautifully. It is very clear, interesting, and not dry despite being mathematically rigorous. Although the author is a mathematician, he wrote this book in a way that makes it accessible to mathematically-oriented graduate students and researchers in computational and applied mechanics and engineering. The basic ideas are described clearly and in a well-organized way. In addition, one can learn quite a lot from this book about areas as diverse as singular integrals, harmonic functions, and domain decomposition. The summary on singular integrals in Section 1.4, for example, is superb. The book, with 540 pages, is divided into seven chapters. Chapter 1 introduces the main ideas of NBR, and is followed by Chapters 2–5, which apply the method to the four differential equations mentioned above. Chapter 6 discusses the coupling of NBR and Finite Elements. This is similar to the DtN method, which has been devised independently in the west primarily for wave problems (which are not discussed at all in this book). The last chapter shows how to solve unbounded domain problems using NBR combined with iterative Domain Decomposition. The book also includes a preface (telling about Prof Feng), a list of more than 200 references (about half of which are Chinese publications), and an index. The NBR method is described on the back cover as a competitor to the standard BEM. Such a comparison is not totally appropriate. The BEM is a general method that can handle problems in irregular geometries. It makes use of the full-space Green’s function associated with the operator involved, which does not depend on the given geometry. On the other hand, NBR makes use of the natural integral operator, which depends on the specific geometry under consideration. Finding this operator on Γ (analytically) is equivalent to solving the problem in Ω analytically. Thus, in its simplest form, the NBR method can be applied in practice only in cases where the analytic solution is already known! Of course, there is no point in using NBR in such a manner. However, this does not mean that the idea underlying NBR is not useful. It is definitely useful when the domain of the problem under consideration involves a region where the problem is analytically-solvable and a region where it is not. Such problems are discussed in the last two chapters, where the practical usefulness of the method is demonstrated. In addition, understanding the various ways in which a problem can be represented (via partial differential equations, as a variational problem, by an integral equation, etc) is always beneficial and may lead to interesting insights regarding the properties of solutions and to new computational methods. Natural Boundary Integral Method and Its Applications is neither a course textbook nor a state-of-the-art research book on a wide scientific area, but it is a satisfying self-contained summary of a very interesting piece of work that has been hidden from the western reader so far. This monograph is highly recommended as an enjoyable and eye-opening reading for the mathematically-oriented researcher and practitioner of applied mechanics.

A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

Nonreflecting boundary condition for the Helmholtz equation