Two-dimensional Boundary Value Problems Research Articles

Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function

A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly.

Steady-State Vibrations of Two-Layer Plates With Rigidly Fixed end Faces and Imperfect Contact of the Layers

We study vibrations of a two-layer plate of any thickness with rigidly fixed plane faces and sliding contact of the layers. By the method of homogeneous solutions, we reduce the initial boundary-value problem to a countable set of two-dimensional boundary-value problems. Methods for the investigation of the spectral characteristics of SH and P-SV waves are proposed. We study the influence of the mechanical and geometric parameters on the variations of the phase and group velocities and the cutoff frequencies.

Generalized Warping Analysis of Composite Beams of an Arbitrary Cross Section by BEM. II: Numerical Applications

AbstractIn this two-paper contribution, a general formulation for the nonuniform warping analysis of composite beams of arbitrary simply or multiply connected cross sections, under arbitrary external loading and general boundary conditions, is presented. In Part I, the governing differential equations have been derived, leading to 10 boundary value problems with respect to the kinematical components and independent warping parameters as well as to seven two-dimensional boundary value problems for the involved shear and torsional warping functions. The aforementioned boundary value problems for the beam are numerically solved using the analog equation method (AEM), a boundary element method (BEM)-based technique. The warping functions and the geometric constants, including the additional ones due to warping, are evaluated with a pure BEM approach. In this Part II, numerical applications of great practical interest are worked out to demonstrate the efficiency and accuracy of the developed method compared wi...

On the multigrid cycle strategy with approximate inverse smoothing

Purpose– The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers.Design/methodology/approach– The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE).Findings– Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis).Research limitations/implications– The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern.Originality/value– A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

Existence of solutions of a two-dimensional boundary value problem for a system of nonlinear equations arising in growing cell populations

In the paper [A. Ben Amar, A. Jeribi, and B. Krichen, Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg's model type, to appear in Math. Slovaca. (2014)], the existence of solutions of the two-dimensional boundary value problem (1) and (2) was discussed in the product Banach space Lp×Lp for p∈(1, ∞). Due to the lack of compactness on L1 spaces, the analysis did not cover the case p=1. The purpose of this work is to extend the results of Ben Amar et al. to the case p=1 by establishing new variants of fixed-point theorems for a 2×2 operator matrix, involving weakly compact operators.

Thermo-Elastic Half Space Under Illumination of a Laser Beam by Using Lord and Shulman Theory

The problem of thermo-elasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a two-dimensional boundary value problem for a semi-infinite medium heated by a pulsed laser beam. The surface of the medium is taken as traction free. The general solution is obtained in the Laplace and Hankel transforms domain. Approximate small time analytical solutions to temperature, displacement, and stress are obtained. Results of this problem are presented graphically.

The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions

We investigate applications of the method of fundamental solutions (MFS) for the numerical solution of two-dimensional boundary value problems in complex geometries, governed by the Laplace equation and subject to Dirichlet boundary conditions which are not harmonic. Such problems can be very challenging because of the appearance of boundary singularities. We consider several ways of choosing the boundary collocation points as well as the source points in the MFS. We show that with an appropriate such choice the MFS yields highly accurate results.

Parallel multigrid algorithms based on generic approximate sparse inverses: an SMP approach

New parallel computational techniques are introduced for the parallelization of Generic Approximate Sparse Inverse multigrid methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Generic Approximate Sparse Inverse Matrix (GenAspI) algorithm is achieved based on a new computational approach, namely "strip," which utilizes the data independence of the rows assigned in each available processor. Additionally, new parallel computational techniques are proposed for the parallelization of a modified multigrid V-Cycle method, based on POSIX Threads, for multicore systems. The modified V-Cycle utilized a Parallel PGenAspI Preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) as a coarse solver to ensure better parallel performance of the multigrid method. For parallelization purposes, a replication of the multigrid method function is executed on each processor with different index bands and with proper synchronization points to ensure less thread-creation overhead and to maximize parallel performance. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PGenAspI algorithm and the PGenAspI---MGV method for solving classical two-dimensional boundary value problems on multicore computer systems are presented. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.

Свободные колебания анизотропных цилиндрических оболочек с переменными параметрами

Investigations of the free vibrations of open orthotropic cylindrical shells with circumferentially varying thickness and the different boundary conditions based on the Mindlin shell theory are presented. The approach is based on reduction of two-dimensional boundary-value problem to one-dimensional using the spline-approximation method along one coordinate. The problem is described by the system of partial equations of 10 th power with the variable coefficients.The one-dimensional eigenvalues problem is solved by the stable numerical discrete orthogonalization method in combination with the step-by-step search method. The calculations for the various boundary conditions on the end-faces and different variants shapes of cylinder thickness for orthotropic cylindrical shells are presented

Wavelet solution of a class of two-dimensional nonlinear boundary value problems

By combining techniques of boundary extension and Coiflet-type wavelet expansion, an approximation scheme for a function defined on a two-dimensional bounded space is proposed. In this Wavelet approximation, each expansion coefficient can be directly obtained by a single-point sampling of the function. And the boundary values and derivatives of the bounded function can be embedded in the modified wavelet basis. Based on this approximation scheme, a modified wavelet Galerkin method is developed for solving two-dimensional nonlinear boundary value problems, in which the interpolating property makes the solution of such strong nonlinear problems very effective and accurate. As an example, we have applied the proposed method to the solution of two-dimensional Bratu-like equations. Results demonstrate an excellent numerical accuracy, a 2.5-order convergence rate for the present wavelet method, and a very good capability in dealing with the nonlinear boundary value problems with multiple solution branches. Interestingly, unlike most existing methods, numerical errors of the present solutions are not sensitive to the nonlinear intensity of the two-dimensional equations.

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work [1–3] from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.

A NOTE ON PARALLEL FINITE DIFFERENCE APPROXIMATE INVERSE PRECONDITIONING ON MULTICORE SYSTEMS USING POSIX THREADS

New parallel computational techniques are introduced for the parallelization of explicit finite difference (FD) approximate inverse matrix methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Optimized Banded Generalized Approximate Inverse Matrix (OBGAIM) algorithm is achieved based on the concept of the "fish bone" approach with the use of a thread pool pattern. Theoretical estimates on speedups and efficiency are also presented. Additionally, new parallel computational techniques are proposed for the parallelization of explicit preconditioned biconjugate conjugate gradient type methods, based on POSIX threads, for multicore systems. For parallelization purposes a replication of the parallel explicit preconditioned biconjugate conjugate gradient-STAB (PEPBICG-STAB) method was assigned on each created thread, with different index bands and with proper synchronization points on inner products and matrix-vector multiplications. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the Parallel Fish Bone OBGAIM (PaFiBo-OBGAIM) algorithm and the PEPBICG-STAB method for solving classical two-dimensional boundary value problems on multicore computer systems are presented, which are favorably compared to corresponding results from multiprocessor systems. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.

Asymptotics of eigenvalues of the two-dimensional Dirichlet boundary-value problem for the Lamé operator in a domain with a small hole

The Dirichlet boundary-value problem for the eigenvalues of the Lame operator in a twodimensional bounded domain with a small hole is studied. The asymptotics of the eigenvalue of this boundary-value problem is constructed and justified up to the power of the parameter defining the diameter of the hole.

Static and Dynamic Problems for Anisotropic Inhomogeneous Shells with Variable Parameters and Their Numerical Solution (Review)

Studies on the static and dynamic deformation of isotropic and anisotropic elastic shell-like bodies of complex shape performed using classical and refined problem statements are reviewed. To solve two-dimensional boundary-value problems and eigenvalue problems, use is made of a nontraditional discrete-continuum approach based on the spline-approximation of the unknown functions of partial differential equations with variable coefficients. This enables reducing the original problem to a system of one-dimensional problems solved with the discrete-orthogonalization method. An analysis is made of numerical results on the distribution of stress and displacement fields and dynamic characteristics depending on the loading and boundary conditions, geometrical and mechanical parameters of elastic bodies. Emphasis is placed on the accuracy of the results

Coupling flat-top partition of unity method and finite element method

We present a novel technique of coupling finite element method with mesh-based flat-top partition of unity method. The proposed coupling method allows us to bind any order of finite elements with flat-top partition of unity method. To verify the coupling, we test the coupling method on one- and two-dimensional boundary value problems including linear elasticity problem on a cracked domain. The coupled formulation provides a platform for stable enrichments to obtain highly accurate solution especially in the enrichment area.

The unsteady flow of a weakly compressible fluid in a thin porous layer III: Three-dimensional computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary-value and eigenvalue problems. In this article, we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: Three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that ϵ ∝ h/l ≪ 1, we show that, to leading order in ϵ, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in ϵ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, ϵ, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

Josephson φ-junctions based on structures with complex normal/ferromagnet bilayer

We demonstrate that Josephson devices with a nontrivial phase difference 0 < φg < π in the ground state can be realized in structures composed of longitudinally oriented normal metal (N) and ferromagnet (F) films in the weak link region. Oscillatory coupling across the F-layer makes the first harmonic in the current–phase relation relatively small, while coupling across the N-layer provides a negative sign of the second harmonic. To derive quantitative criteria for a φ-junction, we have solved the two-dimensional boundary-value problem in the frame of Usadel equations for overlap and ramp geometries of S–NF–S structures. Our numerical estimates show that φ-junctions can be fabricated using up-to-date technology.

Existence Result for a Dynamical Equations of Generalized Marguerre-von Kármán Shallow Shells

In a recent work in the static case, Gratie (Appl. Anal. 81:1107–1126, 2002) has generalized the classical Marguerre-von Karman equations studied by Ciarlet and Paumier in (Comput. Mech. 1:177–202, 1986), where only a portion of the lateral face of the shallow shell is subjected to boundary conditions of von Karman type, while the remaining portion is subjected to boundary conditions of free edge. Then Ciarlet and Gratie (Math. Mech. Solids 11:83–100, 2006) have established an existence theorem for these equations. In Chacha et al. (Rev. ARIMA 13:63–76, 2010), we extended formally these studies to the dynamical case. More precisely, we considered a three-dimensional dynamical model for a nonlinearly elastic shallow shell with a specific class of boundary conditions of generalized Marguerre-von Karman type. Using technics from formal asymptotic analysis, we showed that the scaled three-dimensional solution still leads to two-dimensional dynamical boundary value problem called the dynamical equations of generalized Marguerre-von Karman shallow shells. In this paper, we establish the existence of solutions to these equations using a compactness method of Lions (Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Paris, 1969).

Two-point Taylor approximations of the solutions of two-dimensional boundary value problems

We consider two-dimensional elliptic boundary value problems of the form L(U)=F in Ω⊂R2,Ω bounded and open, with a Dirichlet boundary condition U|∂Ω=H, where L is a second order linear differential operator whose coefficients, as well as the functions F and H are differentiable up to a certain degree. In a recent paper [C. Kesan, Taylor polynomial solutions of second order linear partial differential equations, Appl. Math. Comput. 152 (2004) 29–41], a matrix algorithm is introduced to compute the standard Taylor polynomial of the solution U at a certain point in Ω. We propose an alternative formulation of the problem based on a redefinition of the unknown U and the use of the standard Frobenius method that simplifies the computation of the Taylor coefficients of U. We also consider the use of a two-point Taylor representation of the solution, instead of the classical one-point Taylor representation, which gives a more uniform approximation of the solution.