R. Alsan Meric

Differential and integral sensitivity formulations and shape optimization by BEM

Two different formulations for shape design sensitivity analysis are provided for heat conducting solids. The material derivative concept and the adjoint variable method of analysis are the common techniques utilized in the two formulations. It is found that complementary boundary integral equations arise in the case of the integral sensitivity analysis. Numerical solutions of the primary and adjoint equations are achieved by the BEM, whilst shape configurations are discretized by means of ray and height functions. Several 2-D constrained shape optimization problems are solved by the proposed numerical solution techniques.

Shape design sensitivity analysis and optimization for nonlinear heat and electric conduction problems

Abstract In this article, the shape optimization for the Joule heating of solid bodies is investigated. The conductivity coefficients are permitted to vary with temperature. The shape design sensitivity analysis is performed for a general objective function using Ike adjoint variable method and the material derivative technique. The Kirchhoffs transformation or the Newton-Ralpson method is utilized during the Galerkin finite-element discretizations. The minimization of the objective junction is achieved through the conjugate gradient method. After analytical verifications, two two-dimensional example problems are numerically investigated by the present method.

SIMULTANEOUS OPTIMIZATION OF SHAPE AND FLOW PARAMETERS FOR COMBINED FREE AND FORCED CONVECTION IN VERTICAL CHANNELS

Simultaneous optimization of shape and flow parameters is performed for a combined free and forced convection flow through vertical rectangular channels with moving walls. The laminar flow is assumed to be fully developed in the axial direction. The wall velocity, the axial pressure gradient and the channel height in the transverse plane are taken as the optimization parameters. The sensitivity expressions of both the objective function and the flow rate constraint of optimization are obtained in terms of the relevant physical variables, as well as adjoint variables which satisfy additional p.d.e.s. All equations are discretized using the finite element method. Numerical results are provided for the present constrained optimization problem for various values of the problem parameters which include the moving wall segment size and the Rayleigh number. The results indicate that with increased Rayleigh number the optimal values of the wall velocity and the axial pressure gradient are increased, while the optimal value of the channel height is decreased. General sensitivity expressions are also presented in the appendix which might be utilized for arbitrary boundary variations along with arbitrary optimization objectives in other investigations.

Domain decomposition methods for Laplace's equation by the BEM

Domain decomposition methods for Laplaces equation are investigated in the present paper. An optimization approach is adopted to satisfy the Dirichlet and/or Neumann interface conditions. The adjoint variable method is used to find the sensitivity of the objective function of optimization. The boundary element method is utilized for the discretization of the primary and adjoint problems. Two example problems are studied with the proposed solution method. Comparisons are also provided by using two other techniques, namely the Uzawa and Schwarz methods, already available in the literature. It is also suggested that a parallel implementation of the method could be performed leading to computational efficiency. Copyright

An optimization approach for Stokes flow in a channel with a cylindrical block by the BEM

Abstract The Stokes flow problem in a channel with a cylindrical block is solved by using an optimization approach. The stream function—vorticity formulation is adopted to characterize the flow problem. As the stream function value on the cylindrical block is a priori unknown it is treated as the decision parameter. The objective function to be minimized is taken as the square of the Gauss condition of the vorticity function. The sensitivity analysis of optimization is done by the adjoint variable method, necessitating the solution of an adjoint problem besides the primary (flow) problem. For the discretization of all the equations the boundary element method is utilized. The minimization process is performed by a nonlinear programming method with the convergence within a given tolerance achieved in a few iteration steps. Besides the stream function and vorticity, the pressure distribution is obtained afterwards by solving Laplaces equation. The numerical results are provided for various flow functions in terms of, e.g. streamlines, equivorticity, and constant pressure lines within the flow field.

Shape design sensitivity analysis and optimization for the non-homogeneous Helmholtz equation by the BEM

The present paper deals with the shape design optimization of bodies subjected to a non-homogeneous Helmholtz equation. Using the adjoint variable method the material derivative of a general integral functional is obtained analytically. Boundary integral equations defined only on the boundary are derived using auxiliary fundamental solutions to be used in the boundary element method. Some constrained shape design optimization problems are solved by the proposed numerical procedure.

Boundary elements for joule heating of solids with orthotropic conductivities

A quasi-coupled problem of joule heating in solids has been investigated using the boundary element method. An electric potential field satisfying Poissons equation is first solved within the medium. Internal fluxes of the potential generate an electrically induced heat source. Another Poissons equation representing the heat conduction phenomenon is next solved using the same numerical method. The electrical and thermal conductivities are taken as orthotropic without any space dependency. After verifying the present computer code, four example problems with various conductivities have been analyzed by the proposed solution technique.

An optimization approach for stream function solution of potential flows around immersed bodies

SUMMARYPotential flow problems around immersed bodies have been treated byan optimization approach. When thestream function is used as the field variable, the boundary values may not be known a priori and may betaken asthe decision parametersto minimize integral objective functionals. The circulation integrals aroundthe immersed bodies or the Kutta condition at the trailing edges of the bodies may be used to construct theobjectivefunctionofoptimization.Thesensitivityanalysisneededfortheminimizationprocessisperformedby the adjoint variable method, while the numerical solutions of the primary (flow) and adjoint equationshave been obtained by the finite element method. Having checked the present method with exact solutionsand the classical superposition method, several flow problems involving one or more immersed bodies withor without circulation are investigated numerically. #1998 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng, 14, 253–269 (1998) KEY WORDS potential flow; optimization approach; sensitivity analysis; adjoint variable method; finite elements

Gradient weighted residuals for error indicators in FEM and BEM

A simple yet general approach is presented for checking the adequacy of approximate solutions obtained with FE and BE weighted residual methods. In this method, which is called the gradient weighted residual technique, the computed residuals in the elements are weighted by appropriate gradients and thereby an indicator of the quality of solution is calculated. Such computations are carried out a-posteriori and if need be a reanalysis is then carried out on a new mesh generated on the basis of the quality of the solution determined over the domain.

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.