基于结合扩展精度技术的基本解方法的非线性功能梯度材料热传导问题求解

采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量,将时域问题转化为频域问题求解;采用基本解方法计算得到高精度的频域数值解,再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例,分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点,能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.     

Sensitivity analysis for transient heat conduction in a solid body —Part II: Interface modification

A transient heat conduction problem within a thermal anisotropic multiphase solid body is formulated. An arbitrary thermal response functional defined over space and time domains is introduced and its first-order sensitivities with respect to variation of two kinds of interfaces are discussed. Both variations of shape and material properties of interface are considered. Sensitivity analysis is performed using the direct and adjoint approaches.     

Boundary element analysis of nonlinear transient heat conduction problems

This paper is concerned with a boundary element formulation and its numerical implementation for the nonlinear transient heat conduction problems with temperature-dependent material properties. By using the Kirchhoff transformation for the material properties a set of pseudo-linear integral equations is obtained in space and time for the fully three-dimensional nonlinear problems under consideration. The resulting boundary integral equations are solved by means of the usual boundary element method. Emphasis is placed on the numerical solution procedure employing constant elements with respect to time. It is shown that in this case there is no need to evaluate the domain integrals resulting from the nonlinearity of the problem. Finally, the powerful usefulness of the proposed method is demonstrated through the numerical computation of several sample problems.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

2D and 3D transient heat conduction analysis by BEM via particular integrals

Particular integral formulations are presented for 2D and 3D transient potential flow (heat conduction) analysis. The results of the analysis are compared with an alternative formulation developed using the volume integral conversion approach. Although the mathematical foundation of the two methods are different both formulations are shown to produce almost identical results.For the particular integral formulation, the steady-state heat conduction equation is used as the complementary solution and two global shape functions (GSFs) are considered to approximate the transient term of the heat conduction equation.The numerical results for three example problems are given and compared with their analytical solutions.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

Using computer graphics to teach the heat equation

The graphs of the Fourier series solution to three transient, one-dimensional heat conduction problems were produced. Also, the graphs of the orthogonal function solution to five transient, one-dimensional heat conduction problems with radiating boundary conditions were produced. These were made into two TV tapes which are used in the classroom to help illustrate the mathematical solution to the heat conduction problem.     

Shape design sensitivity analysis and optimization of general shape arches

A method is presented for the shape design sensitivity analysis as applied to general arches whose shapes cannot be mapped on one plane. The shape design sensitivity formulation with respect to the perturbation in the direction normal to the middle surface of the shallow arch curve is derived using the material derivative and the adjoint variable method with the system variational equation expressed in a Cartesian coordinate system. A general shape arch is subdivided into segments each of which can be considered as a shallow arch. On each subdivision of the arch, a Cartesian coordinate system is installed and the shape design sensitivity for the shallow arch is applied. For numerical implementation, the finite element method is adopted and each finite element can be considered as such a subdivision. Numerical examples for sensitivity analysis and optimization are presented to illustrate the finite element sensitivity method proposed.     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.     

An efficient implementation of adjoint sensitivity analysis for optimal control problems

In this paper an efficient implementation of design sensitivity analysis techniques is presented for nonlinear optimal control problems using the adjoint variable method. Techniques for functionals (integrals) and dynamic response (or pointwise) constraints are developed. Emphasis is placed on the proper choice of numerical techniques which exploit the structure of the problem to achieve efficiency. Numerical results for two optimal control examples show great improvement over previous implementations. Unlike previous results the computational effort required for DSA is shown to increase only linearly with the number of discretization points used and is a much smaller percentage of total CPU time.     

Numerical determination of boundary heat fluxes in an enclosure dynamically with natural convection through Fletcher-Reeves gradient method

An iterative Fletcher-Reeves conjugate gradient method (CGM) is adopted to estimate the boundary heat fluxes in a fluid-saturated enclosure, where the fluid flow is dynamically coupled with the heat convection. The sets of direct, sensitivity and adjoint equations required for the solution of the inverse problem are formulated in terms of an arbitrary domain in two dimensions. The methodology of conjugate gradient method solves the inverse natural convection problem satisfactorily without any a priori information about the unknown heat fluxes. The pressure-correction method is utilized to solve the continuum direct, sensitivity and adjoint problems by enforcing global mass and energy conservations. Effects of boundary heat flux profile and thermal Rayleigh number on the convective heat transport are investigated. The effects of position and number of temperature sensors on the inverse problem solution are also addressed in this paper. Inverse solutions of noise data are regularized with the Discrepancy Principle of Alifanov; otherwise, the high frequency components of the random noise were reproduced.     

Sensitivity analysis for transient heat conduction in a solid body —Part I: External boundary modification

A transient hat conduction problem within a thermal anisotropic solid body is formulated. Considering an arbitrary thermal functional defined over space and time domains, its first-order sensitivities with respect to variation of structural material parameters as well as external boundary are derived using the direct and adjoint approaches.     

A study of solution algorithms for shape design sensitivity analysis on a supermini computer with an attached array processor

This paper presents a study and comparison of shape design sensitivity analysis algorithms that are based on the continuum adjoint variable method, the continuum direct differentiation method, and the finite difference method, implemented on a supermini computer with an attached array processor. The basic algorithms and their differences in evaluating shape design sensitivity coefficients are outlined. A solution method for solving a system of equations, using a general sparse storage technique, is used for numerical implementation of shape design sensitivity analysis. It is found that computing shape design sensitivity coefficients using the direct differentiation method is significantly more efficient than using the adjoint variable method or the finite difference method. A detailed performance evaluation of the methods, using an attached array processor, is presented. The performance of the attached array processor, compared to a supermini computer is shown to depend strongly on the type of computations to be carried out. When only parts of a program are running on an attached array processor, the CPU time distribution among the different subroutines of the program can change significantly, compared to using the host processor only.     

基于无网格自然邻接点Petrov-Galerkin法求解带源参数瞬态热传导问题

基于无网格自然邻接点Petrov-Galerkin法,本文建立了一种求解带源参数瞬态热传导问题的新方法.为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻接点插值构造试函数.在局部多边形子域上采用局部Petrov-Galerkin方法建立瞬态热传导问题的积分弱形式.这些多边形子域可由Delaunay三角形创建.时间域则通过传统的两点差分法进行离散.最后通过算例验证了该数值算法的有效性和正确性.     

Parameter and shape sensitivity of thermo-viscoelastic response

Gradient-based optimization methods are still most efficient methods for solving structural optimization problems. The sensitivity formulation has been one of the central issues in the gradient-based optimization algorithm. Thermo-viscoelastic constitutive and parameter sensitivity formulation are presented in this paper. The model considered is composed of two coupled subproblems: the transient heat transfer problem and a rheological, viscoelastic material model known in literature as the standard model. Design variables considered are with material and shape-defining parameters. The investigation includes a finite element formulation and implementation in an object-oriented finite element environment. Results of numerical analysis are presented.     

Transient heat conduction problems using B.I.E.M.

A method, using boundary elements, is presented as a solution to plane transient heat conduction. The proposed method considers the governing equation to be a Helmholtz's equation and solves the problem of time variation using step by step integration. A numerical procedure is developed and its effectiveness verified. Several examples are provided and their results compared with the theoretical ones.     

Thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials

This paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials (FGMs). General formulations are presented based on finite element model by using the direct method and the adjoint method. In the modeling of spatial variances of material properties, the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis. The design variables are the volume fractions of FGMs constituents and structural shape parameters. The design optimization model is then constructed and solved by the sequential linear programming (SLP). Numerical examples are presented to demonstrate the accuracy and the applicability of the present method.     

Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.     

Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows

A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart-Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.