Axisymmetric thermoelastic shape sensitivity analysis and its application to turbine disc design

A general method for shape design sensiti vityi analysis (SDSA) as applied to an axisymmetric thermoelasticity problem is presented using the material derivative concept and the adjoint variable method. The sensitivity of a general functional composed of thermal and mechanical quantities is considered. The method for deriving the sensitivity formula is based on standard direct thermal and elastic boundary integral equation formulation. It is then applied to obtain explicit formulas for a representative displacement and stress constraint imposed on a sector of the boundary. Results of numerical implementation are presented for weight minimization of a turbine disc under thermomechanical loading. The sensitivities of the displacement and stress constraint calculated by the formulas are compared with those by finite differences. Optimum shape obtained under the thermomechanical loading is discussed with that under the mechanical loading only, clearly showing the practical importance of the SDSA of thermoelastic systems.     

Design sensitivity analysis and optimization of interface shape for zoned-inhomogeneous thermal conduction problems using boundary integral formulation

A generalized formulation of the shape design sensitivity analysis for two-dimensional steady-state thermal conduction problem as applied to zoned-inhomogeneous solids is presented using the boundary integral and the adjoint variable method. Shape variation of the external and zone-interface boundary is considered. Through an analytical example, it is proved that the derived sensitivity formula coincides with the analytic solution. In numerical implementation, the primal and adjoint problems are solved by the boundary element method. Shape sensitivity is numerically analyzed for a compound cylinder, a thermal diffuser and a cooling fin problem, and its accuracy is compared with that by numerical differentiation. The sensitivity formula derived is incorporated to a nonlinear programming algorithm and optimum shapes are found for the thermal diffuser and the cooling fin problem.     

THREE-DIMENSIONAL DESIGN SENSITIVITY ANALYSIS USING A BOUNDARY INTEGRAL APPROACH

A general shape design sensitivity analysis approach, different from traditional sensitivity methods is developed for three-dimensional elastostatic problems. The boundary integral design sensitivity formulation is given in order to obtain traction, displacement and equivalent stress sensitivities which are required for design optimization. Those integral equations are derived analytically by differentiation with respect to the normal to the surface at design variable points. Subdivision of boundary elements into sub-elements and rigid body translation methods are employed to deal with singularities that occur during the numerical discretization of the domain. Four different examples are demonstrated to show the accuracy of the method. The boundary integral sensitivity results are compared with the finite difference sensitivity results. Excellent agreement is achieved between the two methods. © 1997 by John Wiley & Sons, Ltd.     

Shape design sensitivity analysis via material derivative-adjoint variable technique for 3-D and 2-D curved boundary elements

A general approach to shape design sensitivity analysis of three- and two-dimensional elastic solid objects is developed using the material derivative-adjoint variable technique and boundary element method. The formulation of the problem is general and first-order sensitivities in the form of boundary integrals for the effect of boundary shape variations are derived for an arbitrary performance functional. Second-order quadrilateral surface elements (for 3-D problems) and quadratic boundary elements (for 2-D problems) are employed in the solution of primary and adjoint systems and discretization of the boundary integral expressions for sensitivities. The accuracy of sensitivity information is studied for selected global performance functionals and also for boundary state fields at discrete points. Numerical results are presented to demonstrate the accuracy and efficiency of this approach.     

Dual boundary element analysis of cracked plates: singularity subtraction technique

The present paper is concerned with the formulation of the singularity subtraction technique in the dual boundary element analysis of the mixed-mode deformation of general homogeneous cracked plates.The equations of the dual boundary element method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation is applied on the other, general mixed-mode crack problems can be solved in a single region boundary element formulation, with both crack surfaces discretized with discontinuous quadratic boundary elements.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional problem unknowns. The single-region boundary element analysis of a general crack problem restricts the availability of singular particular solutions, valid in the global domain of the problem. A modelling strategy, that considers an automatic partition of the problem domain in near-tip and far-tip field regions, is proposed to overcome this difficulty. After the application of the singularity subtraction technique in the near-tip field regions, regularized locally with the singular term of the Williams' eigenexpansion, continuity is restored with equilibrium and compatibility conditions imposed along the interface boundaries. The accuracy and efficiency of the singularity subtraction technique make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

A hyper-singular numerical Green's function generation for BEM applied to dynamic SIF problems

This paper introduces the extension of the numerical Green's function approach for elastodynamic fracture mechanics problems. The formulation uses the hyper-singular boundary integral equation to obtain the fundamental solution for the cracked unbounded medium. The procedure is general and can be applied to multiple crack problems of general geometry. Applications to time harmonic and transient (through inverse numerical Fourier and Laplace transforms) stress intensity factor (SIF) computations are presented and compared with other numerical and analytical results, showing the good accuracy of the present strategy for these kinds of problems.     

Conforming boundary elements in plane elasticity for shape design sensitivity

The structural design sensitivity analysis of a two-dimensional continuum using conforming (continuous) boundary elements is investigated. Implicit differentiation of the discretized boundary integral equations is performed to obtain design sensitivities in an efficient manner by avoiding the factorization of the perturbed matrices. A singular formulation of the boundary element method is used. Implicit differentiation of the boundary integral equations produces terms that contain derivatives of the fundamental solutions employed in the analysis. The behaviour of the singularity of these derivatives of the boundary element kernel functions with respect to the design variables is investigated. A rigid body motion technique is presented to obtain the singular terms in the resulting sensitivity matrices, thus avoiding the problems associated with their numerical integration. A formulation for obtaining the design sensitivities of the continua under body forces of the gravitational and centrifugal types is also presented. The design sensitivity results are seen to be of the same order of accuracy as the boundary element analysis results. Numerical data comparing the performance of conforming and non-conforming formulations in the calculation of design sensitivities are also presented. The accuracy of the present results is demonstrated through comparisons with existing analytical results.     

Structural design sensitivity analysis with general boundary conditions: Dynamic problem

In a previous paper¹, design sensitivity analysis of static response of structural systems when general boundary conditions are imposed during the analysis phase was developed and presented. This paper presents methods for design sensitivity analysis of dynamic response of structural systems when general boundary conditions are imposed during the analysis phase: Eigenvalue as well as transient response problems are discussed. Design sensitivity expression for eigenvalue constraints is derived. For the transient problem, point-wise as well as the integral-type constraint functions are treated. Advantages of these design sensitivity analysis procedures for the dynamic problem are the same as for the static problem. Namely, they are compatible with any existing finite element analysis computer code. Also, they can handle general boundary conditions that are design independent or dependent. Three simple examples are presented to show use of the procedures. Based on these applications, general-purpose numerical procedures can be developed and incorporated into existing computer codes.     

SENSITIVITY ANALYSIS FORMULATION FOR THREE-DIMENSIONAL CONDUCTION HEAT TRANSFER WITH COMPLEX GEOMETRIES USING A BOUNDARY ELEMENT METHOD

A special boundary integral formulation had been proposed to analyse many engineering problems of conduction heat transfer in complex three-dimensional geometries (closely spaced surface and circular hole in infinite domain or simple modification of it) by Rezayat and Burton. One example of such geometries is the mold sets in the injection molding process. In this paper, an efficient and accurate approach for the design sensitivity analysis (DSA) is presented for these kinds of problems in the similar complex geometries using the direct differentiation approach (DDA) based on the above special boundary integral formulation. The present approach utilizes the implicit differentiation of the boundary integral equations with respect to the design variables (radii and locations of circular holes) to yield the sensitivity equations. A sample problem (heat transfer of injection molding cooling system) is solved to demonstrate the accuracy of the present sensitivity analysis formulation. Although the techniques introduced here are applied to a particular problem in heat transfer of injection molding cooling system, their potential application is quite broad.     

Boundary elements in shape design sensitivity analysis and optimal design of vibrating structures

A general approach to shape design sensitivity analysis and optimal design for dynamic transient and free vibrations problems using boundary elements is presented. The material derivatives and the adjoint system method are applied to obtain first-order sensitivities for the effect of boundary shape variations. A numerical example of shape sensitivity analysis and optimal design for free vibrations of an elastic body is presented.     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation

A meshfree approach for plate buckling/post-buckling problems in the case of uniaxial thrust is presented. A geometrical nonlinear formulation is employed using reproducing kernel approximation and stabilized conforming nodal integration. The bending components are represented by Mindlin–Reissner plate theory. The formulation has a locking-free property in imposing the Kirchhoff mode reproducing condition. In addition, in-plane deformation components are approximated by reproducing kernels. The deformation components are coupled to solve the general plate bending problem with geometrical non-linearity. In buckling/post-buckling analysis of plates, the in-plane displacement of the edges in their perpendicular directions is assumed to be uniform by considering the continuity of plating, and periodic boundary conditions are considered in assuming the periodicity of structures. In such boundary condition enforcements, some node displacements/rotations should be synchronized with others. However, the enforcements introduce difficulties in the meshfree approach because the reproducing kernel function does not have the so-called Kronecker delta property. In this paper, the multiple point constraint technique is introduced to treat such boundary conditions as well as the essential boundary conditions. Numerical studies are performed to examine the accuracy of the multiple point constraint enforcements. As numerical examples, buckling/post-buckling analyses of a rectangular plate and stiffened plate structure are presented to validate the proposed approach.     

The dual boundary element method: Effective implementation for crack problems

The present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

Complex potential function in elasticity theory: shear and torsion solution through line integrals

Aim of this paper is to introduce a basis formulation framed into complex analysis valid to solve shear and torsion problems. Solution, in terms of a complex function related to the complete tangential stress field, may be evaluated performing line integrals only. This basis formulation framed into elasticity problems may be a useful support for a boundary method to verify the accuracy of an approximation of function solution. The numerical applications stress the latter point and show the validity of these formulas since exact solutions may be reached for sections where the exact solution is known.     

Kinematic constraints in the state-based peridynamics with mixed local/nonlocal gradient approximations

In contrast to the partial differential equation in the classical continuum mechanics, the equation of motion in standard state-based peridynamics utilizes an integral form and follows an anti-symmetric relationship for the pairwise particle forces. As a consequence, the kinematic constraints such as the boundary displacements and the coupling with other numerical methods in state-based peridynamics cannot be prescribed directly on the geometric boundary for solid mechanics applications. In this paper, an enhanced variant of the state-based peridynamics for the numerical simulation of continuum mechanics problems is presented. The method is first devised based on a convex kernel approximation to localize the influence function on the boundary. A mixed local/nonlocal gradient approximation is introduced to the computation of particle equation of motion and allows a direct imposition of kinematic constraint in the analysis model. The new formulation is shown to retain the conservation nature of state-based peridynamics. Three numerical benchmarks are studied in this paper to demonstrate the effectiveness and accuracy of the proposed method.     

A continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design problems

A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the differentiation of the governing field equations of the direct problem and development of weak forms for the corresponding field sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non‐differentiability of the contact conditions, appropriate regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity fields with respect to various process parameters including the die surface. The accuracy and effectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a Bézier representation of the die curve is introduced. The control points of the Bézier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a finite difference approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the final product or minimize the required extrusion force for a given reduction ratio. An open‐forging die is also designed which for a specified stroke and initial workpiece produces a final product of desired shape. Copyright © 2000 John Wiley & Sons, Ltd.     

Design sensitivity analysis with hypersingular boundary elements

The finite difference load method for shape design sensitivity analysis requires the calculation of stress and stress gradient on the boundary. In the standard boundary element method, the basic state variables-displacement and traction are continuous, and are considered as very accurate. However, the boundary stress and stress gradient, derived from the differentiation of the state variables and Hooke's law, are discontinuous and have relatively lower accuracy than the basic state variables. The hypersingular boundary integral equation is introduced in this paper to determine the stress and stress gradient in the design sensitivity analysis. The numerical examples demonstrate the accuracy of the design sensitivity using the hypersingular boundary elements.     

A numerical study on the elements of shape optimum design

This paper discusses the main elements of shape optimization. The material derivative of a stress function using the continuum approach is derived by introducing an adjoint problem, which is then transformed into shape design sensitivity by replacing the velocity field with the change of the design variables. The difficulty related with the appearance of the concentrated adjoint loads is discussed, with two proposals for the modelling of the adjoint problem. A numerical example is used to demonstrate the accuracy of the proposed formulation for different adjoint loads.

Two shape optimization examples are used to investigate the numerical characteristics of the optimization process. Two kinds of design boundary modelling are employed, namely the linear and cubic spline boundary representation. The difference of the final design shapes under different design variables and mesh distributions are also studied.     

Elastoplastic BIE analysis of cracked plates and related problems. Part 2: Numerical results

Part 1 of this paper reports on the formulation of an advanced boundary—integral equation model for fracture mechanics analysis of cracked plates, subject to elastoplastic behaviour or other, related body force problems. The basis of this formulation contrasts with other BIE elastoplastic formulations in the use of the Green's function for an infinite plane containing a stress free crack. This Green's function formulation assures that the total elastic strain field for the crack problem is accurately imbedded in the numerical model. The second part of this paper reports on the numerical implementation of this algorithm, as currently developed. The anelastic strain field (residual strains, thermal strains, plastic strains, etc.) is approximated as piecewise constant, while the boundary data is modelled with linear interpolations. An iteration solution scheme is adopted which eliminates the need for recalculation of the BIE matrices. The stability and accuracy of the algorithm are demonstrated for an uncracked, notch geometry, and comparison to finite element results is made for the centre-cracked panel. The data shows that even the crude plastic strain model applied is capable of excellent resolution of crack tip plastic behaviour.