High order continuation algorithm and meshless procedures to solve nonlinear Poisson problems

The paper deals with the application of Asymptotic Numerical Method (ANM) for solving non-linear Partial Differential Equations discretized by a meshless technique. In a recent paper [3], it was proposed to associate ANM and Method of Fundamental Solutions (MFS) in a boundary only framework, which permits one to compute a part of non-linear response curves up to the radius of convergence. In the present paper, a continuation algorithm is presented, that is able to compute any solution branch by using the same basis functions. The discretization technique combines fundamental solutions, method of particular solutions (referred as MPS or MFS–MPS when it is coupled with fundamental solutions) and Analog Equation Method (AEM).     

Multi-scale nonlinear modelling of sandwich structures using the Arlequin method

The paper presents a combination of the Arlequin Method (AM) and the Asymptotic Numerical Method (ANM) for studying nonlinear problems related to the mechanical behavior of sandwich composite structures. The Arlequin Method is a multi-scale method in which different models are crossed and glued to each other. The ANM is an alternative method which falls into the category of numerical perturbation techniques. By introducing the power series expansions into the equilibrium equation, the nonlinear problem is transformed into a sequence of linear problems and solved by the standard finite element method. Compared to other classical solvers (Newton–Raphson Method, Modified Newton–Raphson Method), ANM offers a considerable interest in the computation time and reliability. To validate this method, the AM is combined with the ANM to simulate the local damage of 2D–2D and 2D–2D-coupled sandwich beams. The simulation results are compared to a reference solution calculated from a 2D beam without any coupling. In case of the 2D–2D-coupled sandwich beam, the simulation shows a good agreement with the reference solution for both the local damage and the deformation at the loaded point. However, in case of 2D–1D-coupled sandwich beam, the simulation deviate from the reference solution due to the constant thickness of the 1D zig-zag element used to model the 1D zone of the sandwich beam.     

Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models

This work concerns the computation of the nonlinear solutions of forced vibration of damped plates. In a recent work (Boumediene et al. in Comput Struct 87:1508–1515, 2009), a numerical method coupling an asymptotic numerical method (ANM), harmonic balance method and Finite Element method was proposed to resolve this type of problem. The harmonic balance method transforms the dynamic equations to equivalent static ones which are solved by using a perturbation method (ANM) and the finite element method. The numerical results presented in reference (Boumediene et al. in Comput Struct 87:1508–1515, 2009) show that the ANM is very efficient and permits one to obtain the nonlinear solutions with few matrix triangulation numbers compared to a classical incremental iterative method. However, putting a great number of harmonics (6 or greater) into the load vector leads to tangent matrices with a great size. The computational time necessary for the triangulation of such matrices can then be large. In this paper, reduced order models are proposed to decrease the size of these matrices and consequently the computational time. We consider two reduced bases. In the first one, the reduced basis is obtained by the resolution of a classical eigenvalue problem. The second one is obtained by using the nonlinear solutions computed during the first step of the calculus which is realized with the ANM. Several classical benchmarks of nonlinear damped plates are presented to show the efficiency of the proposed numerical methods.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

Homotopy method of fundamental solutions for solving certain nonlinear partial differential equations

In this study, the homotopy analysis method (HAM) is combined with the method of fundamental solutions (MFS) and the augmented polyharmonic spline (APS) to solve certain nonlinear partial differential equations (PDE). The method of fundamental solutions with high-order augmented polyharmonic spline (MFS–APS) is a very accurate meshless numerical method which is capable of solving inhomogeneous PDEs if the fundamental solution and the analytical particular solutions of the APS associated with the considered operator are known. In the solution procedure, the HAM is applied to convert the considered nonlinear PDEs into a hierarchy of linear inhomogeneous PDEs, which can be sequentially solved by the MFS–APS. In order to solve strongly nonlinear problems, two auxiliary parameters are introduced to ensure the convergence of the HAM. Therefore, the homotopy method of fundamental solutions can be applied to solve problems of strongly nonlinear PDEs, including even those whose governing equation and boundary conditions do not contain any linear terms. Therefore, it can greatly enlarge the application areas of the MFS. Several numerical experiments were carried out to validate the proposed method.     

Solving hyperelastic material problems by asymptotic numerical method

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton–Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton–Raphson method for problems involving hyperelastic structures with large strains and instabilities.     

A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

This paper is concerned with a Taylor series–based continuation algorithm, the so-called Asymptotic Numerical Method (ANM). It describes a generic continuation procedure to apply the ANM principle at best, in other words, that presents a high level of genericity without paying the price of this genericity by low computing performances. The way to quadratically recast a system of equation is now part of the method itself, and the way to handle elementary transcendental function is detailed with great attention. A sparse tensorial formalism is introduced for the internal representation of the system, which, when combined with a block condensation technique, provides a good computational efficiency of the ANM. Three examples are developed to show the performance and the versatility of the implementation of the continuation tool. Its robustness and its accuracy are explored. Finally, the potentiality of this method for complex nonlinear finite element analysis is enlightened by treating 2D elasticity problems with geometrical nonlinearities.     

Active network management for electrical distribution systems: problem formulation,benchmark, and approximate solution

With the increasing share of renewable and distributed generation in electrical distribution systems, active network management (ANM) becomes a valuable option for a distribution system operator to operate his system in a secure and cost-effective way without relying solely on network reinforcement. ANM strategies are short-term policies that control the power injected by generators and/or taken off by loads in order to avoid congestion or voltage issues. While simple ANM strategies consist in curtailing temporary excess generation, more advanced strategies rather attempt to move the consumption of loads to anticipated periods of high renewable generation. However, such advanced strategies imply that the system operator has to solve large-scale optimal sequential decision-making problems under uncertainty. The problems are sequential for several reasons. For example, decisions taken at a given moment constrain the future decisions that can be taken, and decisions should be communicated to the actors of the system sufficiently in advance to grant them enough time for implementation. Uncertainty must be explicitly accounted for because neither demand nor generation can be accurately forecasted. We first formulate the ANM problem, which in addition to be sequential and uncertain, has a nonlinear nature stemming from the power flow equations and a discrete nature arising from the activation of power modulation signals. This ANM problem is then cast as a stochastic mixed-integer nonlinear program, as well as second-order cone and linear counterparts, for which we provide quantitative results using state of the art solvers and perform a sensitivity analysis over the size of the system, the amount of available flexibility, and the number of scenarios considered in the deterministic equivalent of the stochastic program. To foster further research on this problem, we make available at http://www.montefiore.ulg.ac.be/~anm/ three test beds based on distribution networks of 5, 33, and 77 buses. These test beds contain a simulator of the distribution system, with stochastic models for the generation and consumption devices, and callbacks to implement and test various ANM strategies.     

Optimality of the method of fundamental solutions

The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is also right-hand side vector dependent. The first part of this work, in [EABE 33(5): 637-43], revealed the close connection between the ECN and the accuracy of the Method of Fundamental Solutions (MFS) for each given problem. In this paper, we show how the ECN can help achieve the problem-dependent quasi-optimal settings for MFS calculations—that is, determining the position and density of the source points. A series of examples on Dirichlet and mixed boundary conditions shows the reliability of the proposed scheme; whenever the MFS fails, the corresponding value of the ECN strongly indicates to the user to switch to other numerical methods.     

Application of method of fundamental solutions for elasto-plastic torsion of prismatic rods

In this paper torsion of prismatic bars considering elastic-plastic material behavior is studied. Based on the Saint-Venant displacement assumption and deformation theory of plasticity for stress-strain relation the boundary value non-linear problem for stress function is formulated. The purpose of our paper is application of method fundamental solution (MFS) and radial basic function (RBF) for solution of this problem. The non-linear torsion problem in plastic region is solved by means of the Picard iteration. Proposed algorithm is based on solution of the linear Poisson equation on each iteration steps.     

Stochastic Boundary Methods of Fundamental Solutions for solving PDEs

A mesh-free Stochastic Boundary Method (SBM) based on randomized versions of the Method of Fundamental Solutions (MFS) is suggested. The randomization is used in the following steps of MFS: (1) the singular source positions are randomly distributed outside the domain, (2) the large system of linear equations for the weights in the expansion over the fundamental solutions is resolved by a randomized SVD method we introduced in [56], or the randomized projection method we developed in [54]. We construct also a new method of stochastic boundary method based on the inversion of the Poisson formula representing the solution in a disc (a sphere, in R³). We present a series of applications of the suggested SBM: we combine SBM with the Random Walk on Spheres and Random Walk on Boundary algorithms which results in methods giving the solution in any set of arbitrary points, without introducing any mesh in the domain. The Laplace, biharmonic, and the system of elasticity equations are involved in our analysis. We present some numerical results and give a brief discussion of the performance of the suggested methods. The numerical experiments carried out for the Laplace and Lamé equations confirm our conclusion that the best results are obtained with the overdetermined systems generated by MFS where the number of source points is considerably smaller than the number of collocation points.     

A quasi-linear technique applied to the method of fundamental solution

This paper proposes the use of a quasi-linear technique for the method of fundamental solution (MFS) to treat the non-linear Poisson-type equations. The MFS, which is a fully meshless method, often deals with the linear and non-linear poisson equations by approximating a particular solution via employing radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the non-homogeneous term in terms of the dependent variable resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method.     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

The time-marching method of fundamental solutions for wave equations

In this paper, a meshless numerical algorithm is developed for the solution of multi-dimensional wave equations with complicated domains. The proposed numerical method, which is truly meshless and quadrature-free, is based on the Houbolt finite difference (FD) scheme, the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). The wave equation is transformed into a Poisson-type equation with a time-dependent loading after the time domain is discretized by the Houbolt FD scheme. The Houbolt method is used to avoid the difficult problem of dealing with time evolution and the initial conditions to form the linear algebraic system. The MPS and MFS are then coupled to analyze the governing Poisson equation at each time step. In this paper we consider six numerical examples, namely, the problem of two-dimensional membrane vibrations, the wave propagation in a two-dimensional irregular domain, the wave propagation in an L-shaped geometry and wave vibration problems in the three-dimensional irregular domain, etc. Numerical validations of the robustness and the accuracy of the proposed method have proven that the meshless numerical model is a highly accurate and efficient tool for solving multi-dimensional wave equations with irregular geometries and even with non-smooth boundaries.     

The quasi-linear method of fundamental solution applied to non-linear wave equations

This paper presents a new meshless method developed by combining the quasi-linear method of fundamental solution (QMFS) and the finite difference method to analyze wave equations. The method of fundamental solution (MFS) is an efficient numerical method for solution Laplace equation for both two- and three-dimensional problems. The method has also been applied for the solution of Poisson equations and transient Poisson-type equations by finding the particular solution to the non-homogeneous terms. In general, approximate particular solutions are constructed using the interpolation of the non-homogeneous terms by the radial basis functions (RBFs). The interpolation in terms of RBFs often leads to a badly conditioned problem which demands special cares. The current work suggests a linearization scheme for the non-homogeneous term in terms of the dependent variable and finite differencing in time resulting in Helmholtz-type equations whose fundamental solutions are available. Consequently, the particular solution is no longer needed and the MFS can be directly applied to the new linearized equation. The numerical examples illustrate the effectiveness of the presented method.     

The MFS–MPS for two-dimensional steady-state thermoelasticity problems

We consider the numerical approximation of the boundary and internal thermoelastic fields in the case of two-dimensional isotropic linear thermoelastic solids by combining the method of fundamental solutions (MFS) with the method of particular solutions (MPS). A particular solution of the non-homogeneous equations of equilibrium associated with a planar isotropic linear thermoelastic material is derived from the MFS approximation of the boundary value problem for the heat conduction equation. Moreover, such a particular solution enables one to easily develop analytical solutions corresponding to any two-dimensional domain occupied by an isotropic linear thermoelastic solid. The accuracy and convergence of the proposed MFS–MPS procedure are validated by considering three numerical examples.     

The method of fundamental solutions for inhomogeneous elliptic problems

We investigate the use of the Method of Fundamental Solutions (MFS) for solving inhomogeneous harmonic and biharmonic problems. These are transformed to homogeneous problems by subtracting a particular solution of the governing equation. This particular solution is taken to be a Newton potential and the resulting homogeneous problem is solved using the MFS. The numerical calculations indicate that accurate results can be obtained with relatively few degrees of freedom. Two methods for the special case where the inhomogeneous term is harmonic are also examined.     

A meshless numerical identification of a sound-hard obstacle

We propose a simple meshless method for detecting a rigid (sound-hard) scatterer embedded in a host acoustic homogeneous medium from scant measurements of the scattered near field. This inverse problem is ill-posed since a solution may not be unique and furthermore, small errors in the input data cause large errors in the output solution. We develop a nonlinear minimization regularized method of fundamental solutions (MFS) for obtaining the numerical solution of the inverse problem in question. Although the MFS is restricted to homogeneous media with constant wavenumber, it is easy to use and simple to implement in higher dimensions. The proposed scheme is tested on several numerical examples and its stability is investigated by inverting measurements contaminated by random noise.     

An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation

For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius R, and another along an offset D to the boundary, where R and D are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths R_k for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length R_k is fully determined by the collocated points and a parameter R or D, such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using R or D. Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.     

ANM for stationary Navier–Stokes equations and with Petrov–Galerkin formulation

This paper deals with the use of the asymptotic numerical method (ANM) for solving non‐linear problems, with particular emphasis on the stationary Navier–Stokes equation and the Petrov–Galerkin formulation. ANM is a combination of a perturbation technique and a finite element method allowing to transform a non‐linear problem into a succession of linear ones that admit the same tangent matrix. This method has been applied with success in non‐linear elasticity and fluid mechanics. In this paper, we apply the same kind of technique for solving Navier–Stokes equation with the so‐called Petrov–Galerkin weighting. The main difficulty comes from the fact that the non‐linearity is no more quadratic and it is not evident, in this case, to be able to compute a large number of terms of the perturbation series. Several examples of fluid mechanic are presented to demonstrate the performance of such a method. Copyright © 2001 John Wiley & Sons, Ltd.