Application of cubic Hermitian algorithms to boundary element analysis of heat conduction

This paper presents the application of cubic Hermitian interpolation based finite element schemes for the time integration of the differential-algebraic system arising in the dual reciprocity boundary element formulation of transient diffusion problems. Weighted residual procedure is used to obtain the desired recurrence relations. Numerical results presented for three representative problems involving different types of boundary conditions amply demonstrate the high accuracy of the cubic Hermitian schemes.     

A boundary element-based inverse-problem in estimating transient boundary conditions with conjugate gradient method

A Boundary Element Method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e. the conjugate gradient method (CGM), is used to solve the Inverse Heat Conduction Problem (IHCP) of estimating the unknown transient boundary temperatures in a multi-dimensional domain with arbitrary geometry. The results obtained by the CGM are compared with that obtained by the standard Regularization Method (RM). The error estimation based on the statistical analysis is derived from the formulation of the RM. A 99 per cent confidence bound is thus obtained. Finally, the effects of the measurement errors to the inverse solutions are discussed. Results show that the advantages of applying the CGM in the inverse calculations lie in that (i) the major difficulties in choosing a suitable quadratic norm, determining a proper regularization order and determining the optimal smoothing (or regularization) coefficient in the RM are avoided and (ii) it is less sensitive to the measurement errors, i.e. more accurate solutions are obtained. © 1998 John Wiley & Sons, Ltd.     

Uniform bicubic B-splines applied to boundary element formulation for 3-D scalar problems

In this work, uniform bicubic B-spline functions are used to represent the surface geometry and interpolation functions in the formulation of boundary-element method (BEM) for three-dimensional problems. This is done as a natural generalization of cubic B-spline curves, introduced by Cabral et al, for two-dimensional problems. Three-dimensional scalar problems, with particular applications to Laplace and Helmholtz equations, are considered.     

A general boundary element method approach to the solution of three-dimensional frictionless contact problems

A direct boundary element method is presented for three-dimensional stress analysis of frictionless contact problems. The isoparametric formulation of the boundary element method is implemented for the general case of contact in the absence of friction, which is limited to linear elastic homogeneous and isotropic materials. An iterative procedure is employed to determine the correct size of the contact zone by finding a boundary solution compatible with the contact condition. The applicability of the procedure is tested by application to three problems of advancing and conforming contact. The computed results are compared with numerical and analytical solutions where possible.     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

A boundary element method for analysis of thermoelastic deformations in materials with temperature dependent properties

In this paper, the boundary element method (BEM) for solving quasi‐static uncoupled thermoelasticity problems in materials with temperature dependent properties is presented. The domain integral term, in the integral representation of the governing equation, is transformed to an equivalent boundary integral by means of the dual reciprocity method (DRM). The required particular solutions are derived and outlined. The method ensures numerically efficient analysis of thermoelastic deformations in an arbitrary geometry and loading conditions. The validity and the high accuracy of the formulation is demonstrated considering a series of examples. In all numerical tests, calculation results are compared with analytical and/or finite element method (FEM) solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

The 3-D thermoelastic boundary element method: semi-analytical integration for subparametric triangular elements

In this paper a semi-analytical integration scheme is described which is designed to reduce the errors that can result with the numerical evaluation of integrals with singular integrands. The semi-analytical scheme can be applied to quadratic subparametric triangular elements for use in thermoelastic problems. Established analytical solutions for linear isoparametric triangular elements are combined with standard quadrature techniques to provide an accurate integration scheme for quadratic subparametric triangular elements. The use of subparametric elements provides an efficient means for coupling thermal and elastostatic analyses. It is possible for the same mesh to be employed, with linear isoparametric elements used for thermal analysis and quadratic subparametric elements used for deformation analysis. Numerical tests are performed on simple test problems to demonstrate the advantages of the semi-analytical approach which is shown to be orders of magnitude more accurate than standard quadrature techniques. Moreover, the expected increased accuracy with subparametric elements is also demonstrated. © 1998 John Wiley & Sons, Ltd.     

Using the boundary element method to solve rolling contact problems

A new approach to steady-state rolling, with and without force transmission, based on the boundary element method is presented. The proposed formulation solves the problem in a more general way than semi-analytical methods, with which it shares some approximations. The robustness and accuracy of the proposed method is reflected in the comparative analysis of the results obtained for three different types of rolling problems involving identical, dissimilar and tyred cylinders, respectively.     

The boundary element method applied to a moving free boundary problem

In this paper a boundary problem is considered for which the boundary is to be determined as part of the solution. A time‐dependent problem involving linear diffusion in two spatial dimensions which results in a moving free boundary is posed. The fundamental solution is introduced and Green’s Theorem is used to yield a non‐linear system of integral equations for the unknown solution and the location of the boundary. The boundary element method is used to obtain a numerical solution to this system of integral equations which in turn is used to obtain the solution of the original problem. Graphical results for a two‐dimensional problem are presented. Published in 1999 by John Wiley & Sons, Ltd.     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Weak-form element differential method for solving mechanics and heat conduction problems with abruptly changed boundary conditions

Element differential method (EDM), as a newly proposed numerical method, has been applied to solve many engineering problems because it has higher computational efficiency and it is more stable than other strong-form methods. However, due to the utilization of strong-form equations for all nodes, EDM become not so accurate when solving problems with abruptly changed boundary conditions. To overcome this weakness, in this article, the weak-form formulations are introduced to replace the original formulations of element internal nodes in EDM, which produce a new strong-weak-form method, named as weak-form element differential method (WEDM). WEDM has advantages in both the computational accuracy and the numerical stability when dealing with the abruptly changed boundary conditions. Moreover, it can even achieve higher accuracy than finite element method (FEM) in some cases. In this article, the computational accuracy of EDM, FEM, and WEDM are compared and analyzed. Meanwhile, several examples are performed to verify the robustness and efficiency of the proposed WEDM.     

The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion

The scaled boundary finite element method, alias the consistent infinitesimal finite element cell method, is developed starting from the diffusion equation. Only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one. No fundamental solution is necessary, and thus no singular integrals need to be evaluated. Essential and natural boundary conditions on surfaces and conditions on interfaces between different materials are enforced exactly without any discretization. The solution of the function in the radial direction is analytical. This method is thus exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions. The semi‐analytical solution inside the domain leads to an efficient procedure to calculate singularities accurately without discretization in the vicinity of the singular point. For a bounded medium symmetric steady‐state stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption. Copyright © 1999 John Wiley & Sons, Ltd.     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

Dual boundary element method for anisotropic dynamic fracture mechanics

In this work, the dual boundary element method formulation is developed for effective modelling of dynamic crack problems. The static fundamental solutions are used and the domain integral, which comes from the inertial term, is transformed into boundary integrals using the dual reciprocity technique. Dynamic stress intensity factors are computed from crack opening displacements. Comparisons are made with quasi‐isotropic as well as anisotropic results, using the sub‐region technique. Several examples are presented to assess the accuracy and efficiency of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Application of the boundary element method to three-dimensional potential problems in heterogeneous media

The present work discusses a solution procedure for heterogeneous media three-dimensional potential problems, involving nonlinear boundary conditions. The problem is represented mathematically by the Laplace equation and the adopted numerical technique is the boundary element method (BEM), here using velocity correcting fields to simulate the conductivity variation of the domain. The integral equation is discretized using surface elements for the boundary integrals and cells, for the domain integrals. The adopted strategy subdivides the discretized equations in two systems: the principal one involves the calculation of the potential in all boundary nodes and the secondary which determines the correcting field of the directional derivatives of the potential in all points. Comparisons with other numerical and analytical solutions are presented for some examples.     

Boundary element analysis in thermoelasticity with and without internal cells

In this paper, a set of internal stress integral equations is derived for solving thermoelastic problems. A jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations. The strongly singular domain integral is then regularized using a semi‐analytical technique. To avoid the requirement of discretizing the domain into internal cells, domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method (RIM). Two numerical examples for 2D and 3D, respectively, are presented to verify the derived formulations. Copyright © 2003 John Wiley & Sons, Ltd.     

A boundary element method for the solution of a class of heat conduction problems for inhomogeneous media

A boundary element method is derived for solving the two-dimensional heat equation for an inhomogeneous body subject to suitably prescribed temperature and/or heat flux on the boundary of the solution domain. Numerical results for a specific test problem is given.     

Microscale heat conduction in homogeneous anisotropic media: a dual-reciprocity boundary element method and polynomial time interpolation approach

In this paper, a dual-reciprocity boundary element method based on some polynomial interpolations to the time-dependent variables is presented for the numerical solution of a two-dimensional heat conduction problem governed by a third order partial differential equation (PDE) over a homogeneous anisotropic medium. The PDE is derived using a non-Fourier heat flux model which may account for thermal waves and/or microscopic effects. In the analysis, discontinuous linear elements are used to model the boundary and the variables along the boundary. The systems of algebraic equations are set up to solve all the unknowns. For the purpose of evaluating the proposed method, some numerical examples with known exact solutions are solved. The numerical results obtained agree well with the exact solutions.     

A three-dimensional shape optimization for transient acoustic scattering problems using the time-domain boundary element method

We develop a three-dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non-uniform rational B-spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time-domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh-based and employs the piecewise-constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient-based non-linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.     

超急速传热过程中热弹性响应的解析分析

考虑超急速传热过程中诱发的热冲击效应,基于L-S广义热弹性理论,建立了温度突变加热条件下热弹性响应的控制方程组。借助于Laplace正逆变换,在适当简化的条件下推导了一维超急速传热问题热弹性响应的解析表达式。通过对温度场、位移场及应力场的解析求解,给出了超急速传热过程中热波和热弹性波在弹性体内的传递规律,并指出在超急速传热条件下,由于热波和热弹性波的相互叠加作用削弱了热作用产生的热冲击效应。