Linear Combination Of Fundamental Solutions Research Articles

A localized collocation solver based on fundamental solutions for 3D time harmonic elastic wave propagation analysis

This paper presents the localized collocation solver based on fundamental solutions to 3D elastic wave propagation analysis. In the proposed collocation solver, the approximated solution at the considered node is represented by a linear combination of the solutions at a few nearest nodes in the stencil support of the considered node instead of the whole discretization nodes. The related weighting functions are determined by solving a system of linear equations, which is constructed as a linear combination of fundamental solutions in the related stencil support. Therefore, the proposed collocation solver produces sparse resultant matrix, which makes it possible to perform large-scale elastic wave propagation simulations on a desktop computer. Furthermore, it avoids highly ill-conditioned dense matrix encountered in most of collocation methods. The efficiency and accuracy of the proposed method have been verified under several benchmark examples.

On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.

On the Green function of an orthotropic clamped plate in a half-plane

First, we calculate, in a heuristic manner, the Green function of an orthotropic plate in a half-plane which is clamped along the boundary. We then justify the solution and generalize our approach to operators of the form (Q(partial ')-a^2partial _n^2)(Q(partial ')-b^2partial _n^2) (where partial '=(partial _1,dots ,partial _{n-1}) and a>0,b>0,ane b) with respect to Dirichlet boundary conditions at x_n=0. The Green function G_xi is represented by a linear combination of fundamental solutions E^c of Q(partial ')(Q(partial ')-c^2partial _n^2),cin {a,b}, that are shifted to the source point xi , to the mirror point -xi , and to the two additional points -frac{a}{b}xi and -frac{b}{a}xi , respectively.

Hybrid Finite Element Analysis of Heat Conduction in Orthotropic Media with Variable Thermal Conductivities

A hybrid finite element method is proposed for the heat conduction analysis with variable thermal conductivities. A linear combination of fundamental solutions is employed to approximate the intra-element temperature field while standard one-dimensional shape functions are utilized to independently define the frame temperature field along the element boundary. The influence of variable thermal conductivities embeds in the intra-element temperature field via the fundamental solution. A hybrid variational functional, which involves integrals along the element boundary only, is developed to link the two assumed fields to produce the thermal stiffness equation. The advantage of the proposed method lies that the changes in the thermal conductivity are captured inside the element domain. Numerical examples demonstrate the accuracy and efficiency of the proposed method and also the insensitivity to mesh distortion.

2.5D singular boundary method for acoustic wave propagation

This study proposes a two-and-a-half-dimensional (2.5D) singular boundary method (SBM) to solve three dimensional (3D) acoustic problems with a constant cross section excited by harmonic point sources. The SBM expands the solution of a problem with a linear combination of fundamental solutions of the governing equation. The singular terms in the fundamental solutions are replaced by the origin intensity factors (OIFs). Nevertheless, the SBM suffers quadratically increasing memory consumption and operation count with respect to the degree of freedoms owing to the fully-populated coefficient matrices. This makes the SBM not applicable in road and rail traffic systems, where the domain of interest always is a few hundred meters long. Taking advantage of the invariant cross section in these problems, the 2.5D method is devised to decompose the 3D problem into a summation of a set of simpler 2D problems, and thus enables the SBM to solve 3D traffic acoustic problems. To recast the results of 3D problems, a new formulation for numerical integration is proposed. The proposed 2.5D SBM is tested to two problems in both full-space and half-space to illustrate its effectiveness and feasibility.

A meshless singular boundary method for elastic wave propagation in 2D partially saturated poroelastic media

This paper presents a boundary meshless method, singular boundary method (SBM), in conjunction with the exponential window method (EWM) to simulate the dynamic response of the 2D partially saturated poroelastic media. The problem is firstly solved in the frequency-domain resulted from the time-domain governing equations via the Fourier transformation. Then the SBM approximates the solutions with a linear combination of fundamental solutions with respect to the source points on the physical boundary. To successfully apply the SBM, two issues are addressed. Firstly, the kernel functions, namely the fundamental solutions of frequency-domain governing equations, are derived based on the eigen-analysis. Secondly, the source singularities of the fundamental solutions are desingularized with analytical formulas. After obtaining the SBM formulation, the time-domain solutions can be obtained by the EWM, where the frequency domain windowing technique is introduced to stabilize the oscillations caused by damping parameters. Four numerical examples are carried out to show the validity and accuracy of the proposed method.

Inverse Heat Conduction Problem in Two-Dimensional Anisotropic Medium

The study proposes a novel mesh-free scheme for a two-dimensional space and tests its efficiency for the inverse heat conduction problem in an anisotropic medium. It aims at determining unknown data at the inaccessible boundary with the help of an approximate solution constructed as a linear combination of fundamental solutions and heat polynomials. The advantage of using this combination lies in the fact that it yields ‘shape functions’ which are in turn solution of the considered equation. The crucial point to note here is that the proposed method yields highly accurate results and the number of collocation points required by the altered scheme are lesser than the standard method of fundamental solutions (MFS). As a direct consequence of the ill-posed problem, a highly ill-conditioned system is obtained, which is solved using regularization. Further, numerical results are provided for two-dimensional convex and non-convex domains to prove that the efficiency of the new scheme is slightly better than the traditional MFS, especially for noisy data.

Reconstruction of a volumetric source domain

Inverse and ill-posed problems which consist of reconstructing the unknown support of a three-dimensional volumetric source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz equations. In the case of constant physical properties, the solutions of these elliptic PDEs are sought as linear combinations of fundamental solutions, as in the method of fundamental solutions (MFS). The unknown source domain is parametrized by the radial coordinate, as a function of the spherical angles. The resulting least-squares functional estimating the gap between the measured and the computed data is regularized and minimized using the lsqnonlin toolbox routine in Matlab. Numerical results are presented and discussed for both exact and noisy data, confirming the accuracy and stability of reconstruction.

Moving pseudo-boundary method of fundamental solutions for nonlinear potential problems

In the method of fundamental solutions (MFS), the solution of a boundary value problem (BVP) is approximated by a linear combination of fundamental solutions expressed in terms of sources which are located on a pseudo–boundary outside the domain of the problem. A major issue in such applications is the optimal placement of these sources. In this work we try to address this issue by letting some parameters which control the location of the sources free, to be determined as well as the coefficients of the fundamental solutions in the MFS approximations. When this approach is applied to nonlinear BVPs, this results in a system of nonlinear equations which can be solved by means of the MATLAB© routines lsqnonlin and fsolve. We investigate the effect of providing the exact Jacobian of the system to these routines to the overall efficacy of the approach. The results of several numerical experiments are presented and analyzed.

The Best Approximation of Plane-Parallel Potential Field on the Set of Fundamental Solutions

The problem of constructing the element of the best approximation of the electric or magnetic plane-parallel field potentials on the set of parametrically fundamental solutions of the Laplace equation in a bounded domain is considered. It is assumed that at the boundary of the region the potentials satisfy the Dirichlet boundary condition or the mixed boundary condition. The best approximation element is sought in the form of a linear combination of fundamental solutions whose singular points are located outside the computational domain. The problem is reduced to finding the conditional minimum of the residual function characterizing the error of the potential approximation on the set of fundamental solutions. It is shown that the gradient descent method can be used to solve the optimization problem. The coordinates of special points of basic fundamental solutions and the coordinates of the linear combination formed by them are used as variable variables to minimize the residual. A numerical algorithm is proposed to find the element of the best approximation, the implementation of which does not require variation of the coordinates of special points. The results of the solution of the test problems are presented, which suggest the possibility of approximating the potential of the plane-parallel field on the set of fundamental solutions with high accuracy even with a small number of basis functions.

Hybrid fundamental solution based finite element method for axisymmetric potential problems with arbitrary boundary conditions

A hybrid fundamental solution based finite element method (HFS-FEM) is proposed to analyze axisymmetric potential problems with arbitrary boundary conditions. The axisymmetric geometry is simplified from the three-dimensional (3D) to the two-dimensional (2D) by expanding boundary conditions into the summation of Fourier series. In the proposed approach, the interior potential field is constructed by utilizing a linear combination of fundamental solutions at source points as intra-element trial functions. And the frame potential field is independently introduced to enforce the continuity between adjacent elements. And then the two assumed fields are expanded into a series of symmetric and asymmetric components as done for the boundary conditions. For each component, the element stiffness equation involving boundary integrals only is established by means of the axisymmetric form of Hellinger-Reissner functional. Finally, the superposition principle is employed for the final solution. To assess the performance of HFS-FEM, three numerical examples are investigated and comparisons are conducted between the proposed approach and ABAQUS. The results show that the HFS-FEM exhibits insensitivity to mesh distortion.

An Algorithm of Linear Combinations: Thermal Conduction

This paper presents computational algorithms that make it possible to overcome some difficulties in the numerical solving boundary value problems of thermal conduction when the solution domain has a complex form or the boundary conditions differ from the standard ones. The boundary contours are assumed to be broken lines (the 2D case) or triangles (the 3D case). The boundary conditions and calculation results are presented as discrete functions whose values or averaged values are given at the geometric centers of the boundary elements. The boundary conditions can be imposed on the heat flows through the boundary elements as well as on the temperature, a linear combination of the temperature and the heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of fundamental solutions of the Laplace equation and their partial derivatives, as well as any solutions of these equations that are regular in the solution domain, and the values of functions which can be calculated at the points of the boundary of the solution domain and at its internal points. If a solution included in the linear combination has a singularity at a boundary element, its average value over this boundary element is considered.

Reconstruction of an elliptical inclusion in the inverse conductivity problem

This study reports on a numerical investigation into the open problem of the unique reconstruction of an elliptical inclusion in the potential field from a single set of nontrivial Cauchy data. The investigation is based on approximating the potential fields of a composite material as a linear combination of fundamental solutions for the Laplace equation with sources shifted outside the solution domain and its boundary. The coefficients of these finite linear combinations are unknown along with the centre, the lengths of the semi-axes and the orientation of the sought ellipse. These are determined by minimizing the least-squares objective functional describing the gap between the given and computed data. The extension of the proposed technique for the reconstruction of two ellipses is also considered.

The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering

We employ the method of fundamental solutions (MFS) for detecting a scatterer surrounding a host acoustic homogeneous medium D due to a given point source inside it. On the boundary of the unknown scatterer (assumed to be star-shaped), allowing for the normal velocity to be proportional to the excess pressure, a Robin impedance boundary condition is considered. The coupling Robin function λ may or may not be known. The additional information which is supplied in order to compensate for the lack of knowledge of the boundary ∂D of the interior scatterer D and/or the function λ is given by the measurement of the scattered field (generated by the interior point source) on a curve inside D. These measurements may be contaminated with noise so their inversion requires regularization. This is enforced by minimizing a penalised least-squares functional containing various regularization parameters to be prescribed. In the MFS, the unknown scattered field us is approximated with a linear combination of fundamental solutions of the Helmholtz operator with their singularities excluded from the solution domain D and this yields the discrete version of the objective functional. Physical constraints are added and the resulting constrained minimization problem is solved using the MATLAB© toolbox routine lsqnonlin. Numerical results are presented and discussed.

Extending the method of fundamental solutions to non-homogeneous elastic wave problems

Two meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.

A mesh-less method to solve electrical resistance tomography forward problem using singular boundary distributed source method

This paper presents a novel mesh-less method called singular boundary distributed source (SBDS) method to solve the electrical resistance tomography (ERT) forward problem. Mesh-less methods are mathematically simple and computationally efficient i.e. same accuracy that of mesh or domain based methods can be achieved with less computation time. The domain boundary is discretized such that the source and field points coincide on the same boundary and the solution is expressed as a linear combination of fundamental solution of governing equation. The singularity appearing due to coinciding of source and field points is addressed using a hybrid approach. The coefficients corresponding to Neumann boundary are evaluated using the improved distributed source method (IBDS) while the coefficients corresponding to Dirichlet boundary conditions are evaluated using inverse interpolation technique. The hybrid SBDS with complete electrode model is formulated for ERT forward problem. The proposed SBDS method has better accuracy and convergence as compared to other mesh-less methods. Especially, the SBDS method solution is not dependent on distributed source radius therefore is stable compared to the IBDS method. The proposed SBDS method is tested with numerical experiments and the performance is compared against boundary discretization methods such as the IBDS and boundary element methods.

Numerical Solutions of Direct and Inverse Stokes Problems by the Method of Fundamental Solutions and the Laplacian Decomposition

In this study, both of direct and inverse Stokes problems are stably and accurately analyzed by the method of fundamental solutions (MFS) and the Laplacian decomposition. In order to accurately resolve the Stokes problem, the Laplacian decomposition is adopted to convert the Stokes equations into three Laplace equations, which will be solved by the MFS, with an augmented boundary condition. To enforce the satisfactions of continuity equation along whole boundary as an augmented boundary condition will guarantee the satisfactions of mass conservation inside the computational domain. The MFS is one of the most promising boundary-type meshless methods, since the time-consuming tasks of mesh generation and numerical quadrature can be avoided as well as only boundary nodes are needed for numerical implementations. The numerical solutions of the MFS are expressed as linear combinations of fundamental solutions of Laplace equation and the sources are located out of the computational domain to avoid numerical singularity. The numerical solutions for velocity components, pressure and their gradient terms can be obtained by simple summation due to the simplicity of the MFS. Several numerical examples of direct and inverse Stokes problems are analyzed by the proposed boundary-type meshless numerical scheme. The simplicity and the accuracy of the proposed method are verified by numerical experiments and comparisons. Moreover, different levels of noise are added into boundary conditions of inverse Stokes problems to validate the stability of the proposed numerical scheme.

The Method of Fundamental Solutions for the Moving Boundary Problem of the One-Dimension Heat Conduction Equation

The melting of the material is regarded as the moving boundary problem of the heat conduction equation. In this paper, the method of fundamental solution is extended into this kind of problem. The temperature function was expressed as a linear combination of fundamental solutions which satisfied the governing equation and the initial condition. The coefficients were gained by use of boundary condition. When the metal wire was melting, process of the moving boundary was gained through the conversation of energy and the Prediction-Correlation Method. A example was given. The numerical solutions agree well with the exact solutions. In another example, numerical solutions of the temperature distribution of the metal wire were obtained while one end was heated and melting.

A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed.

Fundamental Solution Based Finite Element Method

assumed fields (intra-element filed and auxiliary frame field) are employed and the domain integrals in the variational functional can be directly converted to boundary integrals without any appreciable increase in computational effort as in HT-FEM.the assumed intraelement field was constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration, The independent frame field was introduced to guarantee the interelement continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations.