J. H. Kane

Design sensitivity analysis of boundary element substructures

The ability to reduce or condense a three-dimensional model exactly, and then iterate on this reduced size model representing the parts of the design that are allowed to change in an optimization loop is discussed. The discussion presents the results obtained from an ongoing research effort to exploit the concept of substructuring within the structural shape optimization context using a Boundary Element Analysis (BEA) formulation. The first part contains a formulation for the exact condensation of portions of the overall boundary element model designated as substructures. The use of reduced boundary element models in shape optimization requires that structural sensitivity analysis can be performed. A reduced sensitivity analysis formulation is then presented that allows for the calculation of structural response sensitivities of both the substructured (reduced) and unsubstructured parts of the model. It is shown that this approach produces significant computational economy in the design sensitivity analysis and reanalysis process by facilitating the block triangular factorization and forward reduction and backward substitution of smaller matrices. The implementatior of this formulation is discussed and timings and accuracies of representative test cases presented.

Symmetric Galerkin fracture analysis

Abstract The implementation of a symmetric Galerkin boundary integral method for crack problems is described. The symmetric Galerkin procedure requires the derivative (hypersingular) equation on the crack surface, and thus a straightforward application of the dual equation fracture method leads to a non-symmetric matrix. By employing the jump across the fracture as the variable on the crack surface, the problem can be formulated with a symmetric, and smaller dimension, coefficient matrix. An important advantage of this Galerkin approximation is that it avoids the difficulties inherent with C 1 interpolations demanded by a collocation approximation for hypersingular equations. Moreover, by incorporating previously developed efficient analytical integration techniques, the computational cost of this algorithm is shown to be competitive with collocation methods.

Boundary-element shape optimization system for aircraft structural components

The design sensitivities are obtained economically by implicit differentiation of the boundary integral equations. The mesh generation and regeneration is done using a parametric and auxiliary geometry concept that allows the original mesh to remain adequate for a wide range of subsequent evolving geometries as the optimization proceeds

Boundary Formulations for Three-Dimensional Continuum Structural Shape Sensitivity Analysis

The direct, singular, boundary element analysis formulation is shown to provide a basis for a computationally efficient and accurate shape design sensitivity analysis approach for the structural response of three-dimensional solid objects. The theoretical formulation for surface displacement and traction component sensitivities, and all components of the stress tensor is presented along with a formulation for the recovery of displacement and stress components in the interior of the object under consideration. Discussion of computational issues related to the overall efficiency of these formulations is given, along with numerical results to demonstrate the accuracy and efficiency of this approach.

Comparison of shape design sensitivity analysis formulations via material derivative-adjoint variable and implicit differentiation techniques for 3-D and 2-D curved boundary element

This paper reviews and compares two different approaches to continuum structural shape design sensitivity analysis: the material derivative-adjoint variable method and the implicit differentiation method using 3-D and 2-D BEM. Employing identical BEA models, curved element technology, numerical integration and equation solving techniques, the similarities and differences of both methods are highlighted. Significant theoretical and computational aspects associated with both methods are presented and accuracy and computational efficiency characteristics are demonstrated. Advantages and disadvantages of both techniques are identified. The characterization provided can aid in the prudent selection between these sensitivity analysis techniques, based on the characteristics of the proposed application.

Three-dimensional boundary element thermal shape sensitivity analysis

Abstract A computationally efficient and accurate shape design sensitivity analysis (DSA) approach for the thermal response of three-dimensional (3D) solid objects is presented which utilizes a direct, singular, boundary element analysis (BEA) formulation. The theoretical formulations for the primary response (the surface temperature and normal heat flux) sensitivities and the secondary response (tangential components of the heat flux, vector, interior point temperature and heat flux vector components) sensitivities are presented. A number of computational issues related to the overall efficiency of implementation of these formulations are discussed. Numerical results are presented to demonstrate the accuracy and efficiency of this approach.

Boundary element design sensitivity analysis formulations for coupled problems

Abstract Concepts associated with the use of the implicit differentiation technique to derive Boundary Element theoretical formulations for Design Sensitivity Analysis (DSA) of coupled problems are presented. Within the context of a coupled thermal/structural problem, it is shown that the thermoelastic response DSA can be formulated even for transient temperature distributions without the need for explicit domain integrations and without the requirement to factor perturbed matrices. For the case of a coupled structural/fluid problem, a coupled shell structural/fluid acoustic DSA formulation is presented. In this second problem the coupling is twofold. First, there is the obvius coupling of the two physical phenomena governed by different differential equations. Secondly, the shell structural behavior is numerically simulated using the Finite Element Method (FEM) while the fluid acoustic behavior is treated by the employment of the BEM. It is shown that implicit differentiation of the coupled, discretized equations can lead to the formulation of a two step procedure for shape or property DSA for all response quantities involved in the coupled analysis. In both steps of this procedure, it is shown that the DSA approach allows for the reuse of matrix factorizations performed in the earlier coupled analysis step to once again produce a DSA methodology that obviates the need to factor perturbed matrices. These two applications can be considered typical of a much wider class of coupled problems for which effective DSA formulations can be derived via the implicit differentiation approach.

The Influence of Massively Parallel Processing in Boundary Element Computations

Computational efficiencies in executing boundary element elasto-static problems on a massively parallel processor system (Connection Machine) are examined by making comparisons with the corresponding results obtained on a supercomputer (CRAY XMP). The problem chosen is characteristic of the size and type encountered in CPU intensive phases of a typical calculation i.e. calculation of integrals and the solution of the resulting matrix. The surface elements used in these calculations are three dimensional, eight-noded quadrilateral, isoparametric patches. The solver is based on a Block LU decomposition procedure. The results indicate that significant reductions in CPU time can be obtained for those aspects of the procedure that are computationally intensive. Thus, parallel computation of boundary element analysis has the potential to enhance the ability of engineers to solve realistic large size problems economically.

Elasto-plastic boundary element analysis utilizing a zone condensation technique

Abstract The vast majority of mechanical components experience small plastic strains that are (at most) isolated in localized regions. Otherwise these objects simply could not survive. For such components, multi-zone boundary element analysis (BEA) models can be made that separate the overall part into completely elastic zones and zones experiencing elasto-plastic behavior. A boundary element zone condensation technique is shown to facilitate an exact transformation of such problems into a problem of dramatically reduced size. The reduced size nonlinear problem is then shown to be solvable via iterative techniques using a small fraction of the computer resources that is required without condensation. It is shown that the converged solution of the reduced size problem can be subsequently used to compute the expanded response of the overall BEA model, including the zones that were condensed. Numerical examples are presented to quantify the computational efficiency of the zone condensation technique.

Three dimensional boundary formulations for nonlinear thermal shape sensitivities

Implicit differentiation of the discretized boundary integral equations governing the conduction of heat in three dimensional (3D) solid objects, subjected to nonlinear boundary conditions, and with temperature dependent material properties, is shown to generate an accurate and economical approach for the computation of shape sensitivities. The theoretical formulation for primary response (surface temperature and normal heat flux) sensitivities and secondary response (surface tangential heat flux components and internal temperature and heat flux components) sensitivities is given. Iterative strategies are described for the solution of the resulting sets of nonlinear equations and computational performances examined. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantial computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables. A series of nonlinear sensitivity example problems are presented that have closed form solutions. Sensitivities computed using the boundary formulation are shown to be in excellent agreement with these exact expressions.