Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods.     

Practical applications of an integral equation method for the solution of Laplace's equation

An integral equation method for the solution of Laplace's equation, originally proposed for boundary value problems in a single medium, is here extended to problems involving multiple media. The extended method has been used to compute the internal thermal resistance of electric cables and some numerical results are presented.     

Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems

In this work, we use conformal mapping to transform harmonic Dirichlet problems of Laplace’s equation which are defined in simply-connected domains into harmonic Dirichlet problems that are defined in the unit disk. We then solve the resulting harmonic Dirichlet problems efficiently using the method of fundamental solutions (MFS) in conjunction with fast fourier transforms (FFTs). This technique is extended to harmonic Dirichlet problems in doubly-connected domains which are now mapped onto annular domains. The solution of the resulting harmonic Dirichlet problems can be carried out equally efficiently using the MFS with FFTs. Several numerical examples are presented.      

Computation of exterior potential fields by infinite substructuring

A numerical method is presented to evaluate approximate solutions of elliptic partial differential equations outside an arbitrary convex region. The exterior region is conceived as an infinite assembly of geometrically similar finite element cells. A complete set of spatially discretized shape functions is derived for the entire unbounded exterior within which Laplace's equation is required to be satisfied. These functions with the characteristic of outward decay are evaluated by solving a quadratic eigenproblem. The coefficient matrices therein are furnished by the finite element system matrix (which relates the resulting nodal fluxes to the corresponding field strengths) of a typical cell. Elements of that matrix are viewed in the light of the virtual work principle. The energy terms associated with any pair of those basic functions form a convergent geometrical series over a sequence of similarly shaped elements with increasing characteristic dimensions. The exact infinite sum of the virtual work quantities yields the elements of the boundary matrix for the unbounded region. This enables one to carry out the proposed infinite substructuring scheme over the entire infinite collection of those cells. Finally, an expression to estimate the exterior field at an arbitrary point is also presented. The present study has applications to elastostatic, hydrodynamic, electro- and magneto-static protentials in two and three dimensions.     

Robotic Motion Using Harmonic Functions and Finite Elements

The harmonic functions have proved to be a powerful technique for motion planning in a known environment. They have two important properties: given an initial point and an objective in a connected domain, a unique path exists between those points. This path is the maximum gradient path of the harmonic function that begins in the initial point and ends in the goal point. The second property is that the harmonic function cannot have local minima in the interior of the domain (the objective point is considered as a border). This paper proposes a new method to solve Laplace’s equation. The harmonic function solution with mixed boundary conditions provides paths that verify the smoothness and safety considerations required for mobile robot path planning. The proposed approach uses the Finite Elements Method to solve Laplace’s equation, and this allows us to deal with complicated shapes of obstacles and walls. Mixed boundary conditions are applied to the harmonic function to improve the quality of the trajectories. In this way, the trajectories are smooth, avoiding the corners of walls and obstacles, and the potential slope is not too small, avoiding the difficulty of the numerical calculus of the trajectory. Results show that this method is able to deal with moving obstacles, and even for non-holonomic vehicles. The proposed method can be generalized to 3D or more dimensions and it can be used to move robot manipulators.     

A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

The use of approximate functions for the numerical solution of Laplace' equation by an integral equation

In this paper an integral equation method will be outlined to solve Laplace' equation numerically in a finite area S. The method uses either a function which is an approximation of the unknown potential of a particular solution which is only a good approximation in a part of S. The method is also valid if the approximate function is not a solution of Laplace' equation.     

Local and global error estimations in linear structural dynamics

The study discusses the concept of error estimation in linear elastodynamics. Two different types of error estimators are presented. First ‘classical’ methods based on post-processing techniques are discussed starting from a semidiscrete formulation. The temporal error due to the finite difference discretization is measured independently of the spatial error of the finite element discretization. The temporal error estimators are applied within one time step and the spatial error estimators at a time point. The error is measured in the global energy norm. The temporal evolution of the error cannot be reflected. Furthermore the estimators can only evaluate the mean error of the whole spatial domain. As the second scheme local error estimators are presented. These estimators are designed to evaluate the error of local variables in a certain region by applying duality techniques. Local estimators are known from linear elastostatics and have later on been extended to nonlinear problems. The corresponding dual problem represents the influence of the local variable on the initial problem and may be related to the reciprocal theorem of Betti–Maxwell. In the present study this concept is transferred to linear structural dynamics. Because the dual problem is established over the total space–time domain, the spatial and temporal error of all time steps can be accumulated within one procedure. In this study the space–time finite element method is introduced as a single field formulation.     

Reference analysis for Birnbaum-Saunders distribution

In this paper we consider the Bayesian estimators for the unknown parameters of the Birnbaum-Saunders distribution under the reference prior. The Bayesian estimators cannot be obtained in closed forms. An approximate Bayesian approach is proposed using the idea of Lindley and Gibbs sampling procedure is also used to obtain the Bayesian estimators. These results are compared using Monte Carlo simulations with the maximum likelihood method and another approximate Bayesian approach Laplace’s approximation. Two real data sets are analyzed for illustrative purposes.     

Two-scale Dirichlet-Neumann preconditioners for elastic problems with boundary refinements

The present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements. The proposed approach separates the boundary refinements from the interior of the domain by the mortar method, and uses Dirichlet-Neumann preconditioners to solve the corresponding algebraic system. We prove that the simplest Dirichlet-Neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details. Nevertheless, the situation no longer prevails when the refined boundary is clamped. An enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity. Some numerical tests are performed to confirm the analysis, and the tools are extended by the proposition of a quasi-Newton method in the case of nonlinear elasticity. This paper is an extended version of a work presented at the DD16 conference with proofs and complete numerical results.     

基于Abaqus的水下结构声辐射仿真方法

提出一种在Abaqus中直接完成辐射声场仿真的方法,该方法可大大减少流场网格层数.利用无限元的计算结果求解任意远处球形声场的声压;结合弹性球壳的解析解算例验证该方法的正确性,并将该方法拓展到细长型结构中.结果表明：基于Abaqus无限元的水下结构声辐射的计算方法简单、可行;采用有限元与无限元相结合的方法可以大幅度缩小流场区域.     

Numerical solution of Laplace's equation as an integral equation of the first kind

A Fourier approximation method is developed for the simple layer potential reformulation of Laplace's equation. The efficacy of the method is demonstrated in computational examples, and also analyzed theoretically.     

The direct matrix imbedding technique for computing three-dimensional potential flow about arbitrarily shaped bodies

The direct matrix imbedding technique is used to solve Laplace's equation for the velocity potential numerically about arbitrarily shaped bodies with normal gradient boundary conditions in two and three dimensions. The bodies are imbedded in Cartesian grids overlaying relatively large rectangular and box regions. Solutions are obtained only in those parts of the grid necessary for constructing solutions to potential flow problems. An important subclass of these problems, considered in this paper, is ship wave problems in channels. Uniform and stretched Cartesian grids are considered, and solutions are obtained very quickly. Results are presented.     

Evaluating shakedown under proportional loading by non-linear static analysis

A lower bound method for calculating shakedown loads under proportional loading by static non-linear finite element analysis is presented. Stress fields obtained by static analysis and stress superposition are substituted into Melan’s lower bound shakedown theorem. The proposed method is applied to two sample problems: a thick cylinder under internal pressure and a square plate with a central hole under proportional biaxial loading. The results indicate that the method gives accurate lower bound shakedown loads for these problems.     

An integral equation method for the numerical solution of Laplace's equation without Green's function

The common way to establish an integral equation for the solution of Laplace's equation uses the Green's function of the given equation. It will be shown in this paper that an integral equation can also be constructed by using a particular solution of the Laplace equation as the Kernel of the integral equation.     

Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method

In this paper, the extended Kantorovich method proposed by Kerr is further extended to the eigenvalue problem of elastic stability of various rectangular thin plates. By taking advantage of the availability and reliability of state-of-the-art ordinary differential equation solvers, multi-term trial functions have been employed, which is a significant extension to Kerr’s single term approach. As a result, the accuracy is greatly improved and some special problems that a single-term trial function fails to solve are now accommodated. A large number of numerical experiments have been carried out and the computed buckling loads are either exact or more accurate than the best known results in the literature.     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

Comparative analysis of hybrid-trefftz stress and displacement elements

Summary The alternative stress and displacement models of the hybrid-Trefftz finite element formulation for the analysis of linear boundary value problems are derived in parallel form to emphasise the complementary nature of the fundamental concepts they develop from. In the stress model the stresses in the structural domain and the boundary displacements are independently approximated and inter-element stress continuity is enforced explicitly. Conversely, in the displacement model the displacements in the structural domain and the boundary tractions are independently approximated and inter-element linkage is enforced in the form of displacement continuity. In both models the approximation in the domain is constrained to satisfy locally all field equations, a feature typical of the Trefftz method. Duality is used to interpret physically the finite element equations, which are derived from the fundamental relations of elastostatics. Numerical tests are presented to compare the relative performance of the alternative stress and displacement models.     

An iterative method for solving 2D wave problems in infinite domains

This paper presents a new method for solving two-dimensional wave problems in infinite domains. The method yields a solution that satisfies Sommerfeld's radiation condition, as required for the correct solution of infinite domains excited only locally. It is obtained by iterations. An infinite domain is first truncated by introducing an artificial finite boundary (β), on which some boundary conditions are imposed. The finite computational domain in each iteration is subjected to actual boundary conditions and to different (Dirichlet or Neumann) fictive boundary conditions on β.     

Use of fundamental solutions in the collocation method in axisymmetric elastostatics

A method referred to as the fundamental collocation method is applied to problems of axisymmetric elastostatics. In the method the governing field equations are satisfied exactly using fundamental solutions corresponding to concentrated forces while the boundary conditions are satisfied approximately using an overdeterminate collocation technique. Numerical results are given for two stress concentration problems. The paper is concluded by a critical discussion of the merits of the method.