The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

An inverse Stokes problem using interior pressure data

An inverse boundary value problem associated to the Stokes equations in a domain of two dimensions is considered. This problem requires the determination of the unspecified surface fluid velocity, or one of its components, over a part of its boundary by introducing extra interior pressure measurements. The problem is discretised numerically using the boundary element method (BEM) and the resulting ill-conditioned system of linear algebraic equations is solved using the Tikhonov regularisation method, with the choice of the regularisation parameter based on the L-curve criterion. The numerical technique is validated for some test examples with known analytical solutions. The accuracy of the numerical solutions is checked by comparison with their corresponding exact values and an investigation into stability of the numerical solution is undertaken by the addition of random noise into the interior pressure measurements. It is shown that the BEM provides a stable numerical solution of the Stokes problem which converges to the exact solution as the magnitude of error in the interior data decreases.     

The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data

This study investigates the numerical solution of the Laplace and biharmonic equations subjected to noisy boundary data. Since both equations are linear, they are numerically discretized using the Boundary Element Method (BEM), which does not use any solution domain discretization, to reduce the problem to solving a system of linear algebraic equations for the unspecified boundary values. It is shown that when noisy, lower-order derivatives are prescribed on the boundary, then a direct approach, e.g. Gaussian elimination, for solving the resulting discretized system of linear equations produces an unstable, i.e. unbounded and highly oscillatory, numerical solution for the unspecified higher-order boundary derivatives data. In order to overcome this difficulty, and produce a stable solution of the resulting system of linear equations, the singular value decomposition approach (SVD), truncated at an optimal level given by the L-curve method, is employed. © 1998 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

A fast boundary cloud method for 3D exterior electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least‐squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non‐singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright © 2004 John Wiley & Sons, Ltd.     

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

Summary The uniqueness and mathematical stability of the Dirichlet boundary value problem of linear elastostatics is studied. The problem is posed as a set of partial differential equations in terms of displacements and Dirichlet-type of boundary conditions (displacements) for arbitrary bounded domains. Then for the circular interior domain the closed form analytical solution is obtained, using an extended version of the method of separation of variables. This method with corresponding complete solution allows for the derivation of a necessary and sufficient condition for uniqueness. The results are compared with existing energy and uniqueness criteria. A parametric study of the elastic characteristics is performed to investigate the behaviour of the displacement field and the strain energy distribution, and to examine the mathematical stability of the solution. It is found that the solution for the circular element with hourglass-like boundary conditions will be unique for all v ≠ 0.5, 0.75, 1.0 and will be mathematically stable for all v ≠ 0.75. Locking of the circular element occurs for v = 0.75 as the energy tends to infinity.     

A fast boundary cloud method for exterior 2D electrostatic analysis

An accelerated boundary cloud method (BCM) for boundary‐only analysis of exterior electrostatic problems is presented in this paper. The BCM uses scattered points instead of the classical boundary elements to discretize the surface of the conductors. The dense linear system of equations generated by the BCM are solved by a GMRES iterative solver combined with a singular value decomposition based rapid matrix–vector multiplication technique. The accelerated technique takes advantage of the fact that the integral equation kernel (2D Green's function in our case) is locally smooth and, therefore, can be dramatically compressed by using a singular value decomposition technique. The acceleration technique greatly speeds up the solution phase of the linear system by accelerating the computation of the dense matrix–vector product and reducing the storage required by the BCM. Copyright © 2002 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Analysis of frictional contact problem using boundary element method and domain decomposition method

In this paper, we study the bilateral or unilateral contact with Coulomb friction between two elastic solids, using a domain decomposition method coupled with the boundary element method. The decomposition method we have selected is the Schur complement method, a non‐overlapping technique. It enables to reduce the solution of the global problem to the solution of a problem defined only on the contact surface. Moreover, its principal advantage is that computing is done separately on each solid. We have chosen to associate it with the boundary element method. Indeed, it only requires the discretization of the boundaries of solids. This technique of coupling reduces the number of unknowns and the time of computing. We have applied it to the study of indentation of an elastic foundation by an elastic flat punch and a sphere. In this last case, our results are in conformity with the Hertz theory and the analytical solution of Spence. Moreover, we have shown the influence of friction on the size of the contact radius and on the normal pressure at centre. Copyright © 1999 John Wiley & Sons, Ltd.     

A fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems. Part 2: Neumann boundary conditions

We present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound‐hard scatterer. We apply this method to the solution of a mock‐up submarine problem, and highlight its computational advantages and intrinsic parallelism. A key component of our method is an original idea for addressing a Neumann boundary condition in the general framework of a fictitious domain method. This idea is applicable to many other linear partial differential equations besides the Helmholtz equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Thermomechanical analysis of viscoelastic solids

This paper is concerned with the development of a computational algorithm for the solution of the uncoupled, quasi-static boundary value problem for a linear viscoelastic solid undergoing thermal and mechanical deformation. The method evolves from a finite element discretization of a stationary value problem, leading to the solution of a system of linear integral equations determining the motion of the solid. An illustrative example is included.     

Algebraic multilevel iteration method for lowest order Raviart–Thomas space and applications

An optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems, that have their weak formulation in the space H(div), is presented. The algorithm is developed for the discrete problem obtained by using the lowest‐order Raviart–Thomas space. The method is theoretically analyzed and supporting numerical examples are presented. Furthermore, as a particular application, the algorithm is used for the solution of the discrete minimization problem which arises in the functional‐type a posteriori error estimates for the discontinuous Galerkin approximation of elliptic problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical modeling of projectile penetration into compressible rigid viscoplastic media

We develop computational methods for modeling penetration of a rigid projectile into porous media. Compressible rigid viscoplastic models are used to capture the solid–fluid transition in behavior at high strain rates and account for damage/plasticity couplings and viscous effects that are observed in geological and cementitious materials. A hybrid time discretization is used to model the non‐stationary flow of the target material and the projectile–target interaction, i.e. an explicit Euler method for the projectile equation and a forward (implicit) method for the target boundary value problem. At each time step, a mixed finite element and finite‐volume strategy is used to solve the ‘target’ boundary value problem. Specifically, the non‐linear variational inequality for the velocity field is discretized using the finite element method while a finite‐volume method is used for the hyperbolic mass conservation and damage evolution equations. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented Lagrangian method is adopted. Numerical simulations of penetration into concrete were performed. By conducting a time step sensitivity study, it was shown that the numerical model is robust and computationally inexpensive. For the constants involved in the model (shear and volumetric viscosities, cut‐off yield limit, and exponential weakening parameter for friction) that cannot be determined from data, a parametric study was performed. It is shown that using the material model and numerical algorithms that developed the evolution of the density changes around the penetration tunnel, the shape and location of the rigid/plastic boundary, the compaction zones, and the extent of damage due to air‐void collapse are described accurately. Copyright © 2007 John Wiley & Sons, Ltd.     

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time‐dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first‐order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non‐reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high‐order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

Laplacian decomposition and the boundary element method for solving Stokes problems

The solution of the equations governing the steady incompressible slow viscous fluid flow is analysed using a novel technique based on a Laplacian decomposition instead of the more traditional approaches based on the biharmonic streamfunction formulation or the velocity-pressure formulation. This results in the need to solve the Laplace equations for the pressure and other auxiliary harmonic functions which arise from the ideas of Almansi's decomposition. These equations, which become coupled through the boundary conditions, are numerically solved using the boundary element method (BEM). Results both on the boundary and inside the solution domain are presented and discussed for a simple benchmark test example and a few applications in smooth and non-smooth geometries in order to illustrate that the Laplacian decomposition in combination with BEM provides an efficient technique, in terms of accuracy and convergence, to investigate numerically a Stokes flow.     

A method for linear elasto-static analysis of multi-layered axisymmetrical bodies using Hankel's transform

 A method to determine the distribution of stresses and displacements in an infinite, linear, elastic, multi-layered medium subjected to static axisymmetric loading is presented in this work. By using axisymmetric governing equations, Hankel's transform and matrix analysis, the methodology gives a clearly arranged way to calculate the stresses and displacements in the medium. A numerical method for Hankel's transform is employed to perform the calculation. Two representative examples are studied. The results can be utilized as a fundamental solution for boundary element methods for the linear, elasto-static, axisymmetric multi-layered problem with a little modification.