A direct numerical method for the solution of field problems

A numerical method for the analysis of field problems is described. The algorithm is based upon the generalized Betti-Maxwell theorem. Using a set of known solutions to problems with similar boundary conditions produces a set of ‘integral’ equations for the required solution. Using any convenient numerical integration formula reduces the problem to the solution of a set of simultaneous algebraic equations. The accuracy of the solution depends upon the accuracy of the integration formula as applied to the problem under consideration and is independent of the known auxiliary solutions. The method is described in detail as applied to harmonic problems.     

On the numerical solution of phase change problems in transient non-linear heat conduction

A finite difference solution for the transient non-linear heat conduction with phase change in a finite slab is proposed. A two-time level implicit method is used while Taylor's forward projection method is employed for taking into account the non-linearities. The stability due to the boundary conditions at the moving front is verified. The numerical solution is compared with the analytical solutions and found to be in close agreement.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

A fixed grid finite element method for nonlinear diffusion problems with moving boundaries

A comprehensive formulation for a class of diffusion problems with non-linear conductivities is derived by unifying and combining the freezing index and Kirchhoff transformation concepts. The transformed equations have appropriate continuity characteristics across the unknown moving boundary. The applicability of the fixed grid algorithm for the total solution domain is, accordingly, demonstrated. Associated finite element formulations and solution procedures for the transformed equations are detailed. In addition, selected numerical results for single and two phase Stefan type problems as well as fluid flow in a prescribed cavity are presented for solution verification and illustration.     

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n ? 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi‐infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

A method of numerical solution of one-dimensional stefan problems of two types

A one-dimensional conjugate problem of heat transfer with phase transitions of two types (with interfaces of two phases and with a two-phase zone) is solved by a finite difference method based on the general initial heat conduction equation written with the Dirac delta function. A calculating scheme is developed using a nonuniform spatial net with floating nodes and the method of oppositely directed pivots.Nizhnii Novgorod Polytechnic Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No. 3, pp. 332–340, September, 1993.     

A numerical method for treating indentation problems

Summary Abel transforms are used to simplify the equations describing the deformation of an elastic medium by a rigid indentor with a circular cross-section. The equations derived constitute a convenient starting point for the numerical solution of such problems. As a particular example the case of a parabolic rough punch is treated.     

Boundary integral equation solution of moving boundary phase change problems

Boundary integral equation methods are presented for the solution of some two-dimensional phase change problems. Convection may enter through boundary conditions, but cannot be considered within phase boundaries. A general formulation based on space-time Green's functions is developed using the complete heat equation, followed by a simpler formulation using the Laplace equation. The latter is pursued and applied in detail. An elementary, noniterative system is constructed, featuring linear interpolation over elements on a polygonal boundary. Nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution. The system performs well against basic analytical solutions, using these values in the interphase jump condition, with the simplest formulation of the surface normal at boundary vertices. Because the discretized surface changes automatically to fit the scale of the problem, the method appears to offer many of the advantages of moving mesh finite element methods. However, it only requires the manipulation of a surface mesh and solution for surface variables. In some applications, coarse meshes and very large time steps may be used, relative to those which would be required by fixed grid domain methods. Computations are also compared to original lab data, describing two-dimensional soil freezing with a time-dependent boundary condition. Agreement between simulated and measured histories is good.     

Use of the method of green functions for numerical solution of multidimensional stefan problems

It is shown that use of an auxiliary Green function, namely, the Green function of the boundary-value problem for the Laplace operator with a condition of the third kind artificially introduced on part of the boundary, makes it possible to find numerical solutions of multidimensional Stefan problems with any boundary conditions. The efficiency of the method is verified for a Stefan problem that has an exact solution. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 5, pp. 910–916, September–October, 1998.     

On the numerical solution of linear exterior problems via the uncoupling method

The application of the uncoupling of boundary integral and finite element methods to solve exterior boundary value problems in R ² yields a weak formulation that contains only one boundary term. This is the so-called uncoupling term, which is determined by the boundary integral operator of the single-layer potential acting on a circle centered at the origin. The purpose of this paper is to provide a suitable formula, which combines analytical and numerical methods, to approximately integrate the uncoupling term to any exacteness. Our method provides sharper error estimates than the one that uses Truncated Infinite Fourier Series (TIFE). As a model we consider the exterior Dirichlet problem for the Laplacian, and use linear finite elements for the corresponding Galerkin scheme. Some numerical experiments are also presented. © 1998 John Wiley & Sons, Ltd.     

A numerical solution of axisymmetric problems in elastodynamics

The equations governing the axisymmetric dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary value problems in elastodynamics is presented. Two examples are considered.     

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.     

A new method for the numerical solution of integral equation approximations

A new numerical technique for solving the Ornstein-Zernike equation is described. It is particularly useful in solving the Ornstein-Zernike equation for approximations and pair potentials (such as the Percus-Yevick and mean spherical approximations for finite ranged potentials) which imply a finiteranged direct correlation function since for such approximations the numerical technique is essentially exact. The only approximation involved in such cases is the discretization of direct and total correlation functions over the finite range on which the direct correlation function is nonzero. Thus, the new method avoids truncation of the total correlation function and should permit the critical point and spinodal curve to be mapped out with greater accuracy than is permitted by existing methods. Preliminary explorations on the stability and accuracy of the method are described.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

A second order cone complementarity approach for the numerical solution of elastoplasticity problems

In this paper we present a new approach for solving elastoplastic problems as second order cone complementarity problems (SOCCPs). Specially, two classes of elastoplastic problems, i.e. the J ₂ plasticity problems with combined linear kinematic and isotropic hardening laws and the Drucker-Prager plasticity problems with associative or non-associative flow rules, are taken as the examples to illustrate the main idea of our new approach. In the new approach, firstly, the classical elastoplastic constitutive equations are equivalently reformulated as second order cone complementarity conditions. Secondly, by employing the finite element method and treating the nodal displacements and the plasticity multiplier vectors of Gaussian integration points as the unknown variables, we obtain a standard SOCCP formulation for the elastoplasticity analysis, which enables the using of general SOCCP solvers developed in the field of mathematical programming be directly available in the field of computational plasticity. Finally, a semi-smooth Newton algorithm is suggested to solve the obtained SOCCPs. Numerical results of several classical plasticity benchmark problems confirm the effectiveness and robustness of the SOCCP approach.     

A new numerical method for transient thermal stress problems

A new numerical method of calculating unsteady-state heat conduction and thermal stresses by using improved coefficient matrix patterns on the nodal points of a structure is given. Good agreement is obtained between our results and the rigorous results, and this method seems to yield better accuracy than other well-known approximate methods for two-dimensional transient thermal stress problems.     

A method for the numerical solution of a class of boundary value problems governed by second order elliptic systems

A method for the numerical solution of a class of problems governed by a system of second order elliptic partial differential equations is derived. The solution to the boundary value problem is obtained in terms of an integral taken round part of the boundary of the region under consideration. Some numerical examples are considered and the results obtained are shown to be in excellent agreement with those obtained either analytically or by employing other numerical methods.