Boundary Value Problems Of Physics Research Articles

Meshfree computation of electrostatics and related boundary value problems

We discuss a meshfree method for solving boundary value problems in physics. The method uses random data interpolation and radial basis functions that depend on the distance, instead of the usual basis functions that depend on the position. As an example, we apply this method to the electrostatic problem of two parallel plates inside a conducting cylinder and discuss its convergence and robustness. The method is shown to be robust and flexible and can be an efficient alternative to mesh-based methods for solving computational problems involving irregular domains.

A wavelet integral collocation method for nonlinear boundary value problems in physics

A high-order wavelet integral collocation method (WICM) is developed for general nonlinear boundary value problems in physics. This method is established based on Coiflet approximation of multiple integrals of interval bounded functions combined with an accurate and adjustable boundary extension technique. The convergence order of this approximation has been proven to be N as long as the Coiflet with N−1 vanishing moment is adopted, which can be any positive even integers. Before the conventional collocation method is applied to the general problems, the original differential equation is changed into its equivalent form by denoting derivatives of the unknown function as new functions and constructing relations between the low- and high-order derivatives. For the linear cases, error analysis has proven that the proposed WICM is order N, and condition numbers of relevant matrices are almost independent of the number of collocation points. Numerical examples of a wide range of nonlinear differential equations in physics demonstrate that accuracy of the proposed WICM is even greater than N, and most interestingly, such accuracy is independent of the order of the differential equation to be solved. Comparison to existing numerical methods further justifies the accuracy and efficiency of the proposed method.

Comparison of meshfree and mesh-based methods for boundary value problems in physics

Meshfree methods using basis expansion are common in computational physics, but their efficiency and accuracy depend on the quality of basis sets. We describe the application of radial basis functions (RBF), a class of basis functions attracting considerable interest and showing promising results. In the RBF approach, the solutions are interpolated over a chosen set of RBFs scattered over the domain. The value of a given RBF depends on the distance to the center, in contrast to the usual approach where the basis functions depend on the position. The advantage of the RBF method is that it can be efficient, simple to implement, and easily adaptable to irregular domains or higher dimensions. We study the application of this meshfree method to boundary value problems, including electrostatics and quantum systems. We show that the method is robust and flexible, and can be a compelling alternative to mesh-based methods.

Mechanical cloak design by direct lattice transformation

Spatial coordinate transformations have helped simplifying mathematical issues and solving complex boundary-value problems in physics for decades already. More recently, material-parameter transformations have also become an intuitive and powerful engineering tool for designing inhomogeneous and anisotropic material distributions that perform wanted functions, e.g., invisibility cloaking. A necessary mathematical prerequisite for this approach to work is that the underlying equations are form invariant with respect to general coordinate transformations. Unfortunately, this condition is not fulfilled in elastic-solid mechanics for materials that can be described by ordinary elasticity tensors. Here, we introduce a different and simpler approach. We directly transform the lattice points of a 2D discrete lattice composed of a single constituent material, while keeping the properties of the elements connecting the lattice points the same. After showing that the approach works in various areas, we focus on elastic-solid mechanics. As a demanding example, we cloak a void in an effective elastic material with respect to static uniaxial compression. Corresponding numerical calculations and experiments on polymer structures made by 3D printing are presented. The cloaking quality is quantified by comparing the average relative SD of the strain vectors outside of the cloaked void with respect to the homogeneous reference lattice. Theory and experiment agree and exhibit very good cloaking performance.

Solving Integral Equations on Piecewise Smooth Boundaries Using the RCIP Method: A Tutorial

Recursively compressed inverse preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in two-dimensional domains with corners. In a series of subsequent papers, the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: first, to review the RCIP method in a simple setting; second, to show how easily the method can be implemented in MATLAB; third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.

Separable boundary-value problems in physics

Part I Preliminaries 1. Introduction 2. General Theory Part II Two-Dimensional Coordinate Systems 3. Rectangular Coordinates 4. Circular Coordinates 5. Elliptic Coordinates 6. Parabolic Coordinates Part III Three-Dimensional Coordinate Systems 7. Rectangular Coordinates 8. Circular Cylinder Coordinates 9. Elliptic Cylinder Coordinates 10. Parabolic Cylinder Coordinates 11. Spherical Polar Coordinates 12. Prolate Spheroidal Coordinates 13. Oblate Spheroidal Coordinates 14. Parabolic Rotational Coordinates 15. Conical Coordinates 16. Ellipsoidal Coordinates 17. Paraboloidal Coordinates Part IV Advanced Formulations 18. Differential Geometric Formulations 19. Quantum-mechanical Particle Confined to Neighborhood of Curves 20. Quantum-mechanical Particle Confined to Surfaces of Revolution 21. Boundary Perturbation Theory Appendices A Hypergeometric Functions B Baer Functions C Bessel Functions D Lame Functions E Legendre Functions F Mathieu Functions G Spheroidal Wave Functions H Weber Functions I Elliptic Integrals and Functions Index

On configurational forces in short-time dynamics and their computation with an explicit solver

A theoretical description and a computational method to calculate configurational forces in the context of the finite element (FE) method is presented. With respect to problems in short-time dynamics, the fully 3D-case and large deformations in hyper-elastic materials are taken into account. The FE implementation and numerical analysis of different structures demonstrates the applicability of this field of mechanics. In the chosen derivation, the Lagrangian depends on the deformation gradient and on the position (in the reference configuration) explicitly, which accounts for inhomogeneous materials, e.g. materials with phase boundaries, voids or cracks. Analogous to the local balance of linear momentum, the so-called Eshelby stress satisfies a configurational force balance (balance of momentum for the material motion problem), where configurational (or material) forces appear as volume forces in the physical space. A consistent FE description is obtained by formulating the weak form of the configurational force balance. Thus, the configurational forces acting on the finite element nodes may be computed after the physical boundary value problem is solved. For the static case and small deformations, the configurational force is strongly related to the well known J-integral in fracture mechanics.

On sparse lu factorization procedures for the solution of parabolic differential equations in three space dimensions

The numerical solution of partial differential equations in 3 dimensions by finite difference methods leads to the problem of solving large order sparse structured linear systems. In this paper, a factorization procedure in algorithmic form is derived yielding direct and iterative methods of solution of some interesting boundary value problems in physics and engineering.

Boundary Value Problems in Physics and Engineering. By F. Chorlton. Pp. 250. £3.50 1969. (Van Nostrand Reinhold.)

Boundary Value Problems in Physics and Engineering. By F. Chorlton. Pp. 250. £3.50 1969. (Van Nostrand Reinhold.)

Boundary Value Problems

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

The solution of linear boundary value problems in physics by means of the Laplace transformation

The solution of linear boundary value problems in physics by means of the Laplace transformation

The solution of linear boundary-value problems in physics by means of the Laplace transformation

The solution of linear boundary-value problems in physics by means of the Laplace transformation