Krzysztof Dems

SENSITIVITY ANALYSIS IN THERMAL PROBLEMS—II:STRUCTURE SHAPE VARIATION

Abstract The variation of an arbitrary thermal functional corresponding to the variation of structural shape is derived for a thermal isotropic body by using direct and adjoint approaches. Both the steady-state and transient cases are considered, and a simple example is used to demonstrate the various approaches.

SENSITIVITY ANALYSIS IN THERMAL PROBLEMS—I: VARIATION OF MATERIAL PARAMETERS WITHIN A FIXED DOMAIN

For a thermal isotropic body, the first and second variations of an arbitrary thermal functional corresponding to variation of material parameters is derived by using both direct and adjoint approaches. Both the steady-state and transient cases are considered, and the analysis is equally applicable to analytical or numerical solution techniques. A simple example is used to demonstrate the various approaches.

SENSITIVITY ANALYSIS AND OPTIMAL DESIGN FOR STEADY CONDUCTION PROBLEM WITH RADIATIVE HEAT TRANSFER

ABSTRACT A steady heat conduction problem is considered, that is described by the heat conduction equation and the thermal boundary conditions (i.e., Dirichlet, Neumann, Henkel, and radiation conditions on the external boundary, and radiation condition on the hole boundary). An arbitrary behavioral functional is defined and its first-order sensitivity is derived using both the direct and the adjoint approaches. The shape optimization problem is next formulated and two optimization functionals are discussed. The simple numerical example is presented.

Two approaches to the optimal design of composite flywheels

In this article, two approaches to the design of reinforced composite flywheels are presented. The main goal of the optimization procedure is to maximize the accumulated kinetic energy of a flywheel. The first approach is based on a discrete model of reinforcement, causing the discontinuity of static fields along reinforcement and preserving the continuity of kinematic fields. In the second approach, the material of the reinforced flywheel is subjected to the homogenization procedure using the Halpin–Tsai assumption and the continuity of both static and kinematic fields is preserved within the flywheel domain. The evolutionary algorithm was used in both cases to determine the optimal shape of reinforcements, while the finite-element method was applied in order to analyse the mechanical response of a flywheel.

2D shape optimization with static and dynamic constraints

The paper presents an approach that allows us to consider in the shape optimization several static loading conditions together with constraints imposed on eigenfrequencies. The idea of the method is based upon simultaneous solutions of equations and inequalities arising from the Kuhn-Tucker necessary conditions for an optimum problem. The paper is illustrated with four examples in which stress and eigenfrequency are active constraints.

Optimal design of a truss configuration under multiloading conditions

The paper deals with the optimum structural design of a truss for which coordinates of structural nodes as well as member sizes constitute a set of design variables. The truss may be loaded by as many loading conditions as needed and is subjected to constraints imposed on stress displacements and complementary energy. An important relation between the cost function, Lagrange multipliers and limit values of constraints is derived. The paper presents the outline of an algorithm for the solution of a system of equations and inequality arising from the Kuhn-Tucker necessary condition for an optimum problem. Five numerical examples of 2D trusses are presented.

Modeling of Fiber-Reinforced Composite Material Subjected to Thermal Load

Two-dimensional thermally loaded structural elements are considered. The elements are made of composite materials in the form of a multi-layer laminate of matrix layers filled with the fibers. In each layer the fraction of fibers and their arrangement in a matrix can be different. The modeling process of thermal properties of such materials, defined by the coefficients of thermal conductivity in the direction of orthotropy axes (fibers and direction perpendicular to them) is discussed. Using the heat balance equation and Fouriers basic law for total balance of heat flux for mixture of fibers and matrix, the substitute conductivity coefficients in orthotropy directions are determined for a variety of cross-sections of filling fibers in particular layer. These coefficients define the macroscopically equivalent homogeneous orthotropic materials of each layer. Next, the substitute conductivity coefficients for the stack of layers are determined, where the layers stack is modeled with one homogenous material. The proposed model of fiber-reinforced laminate was used in some numerical examples. To analyze the problem of heat transfer within a structure domain, the finite element method was applied.

Second-Order Sensitivities with Respect to Shape of Loaded and Supported Structural Boundaries∗

Abstract For an arbitrary stress, strain, and displacement functional, second-order sensitivities with respect to varying structural shape are discussed. It is assumed that the external boundary of a structure can undergo shape modification described by a set of shape design parameters. Second derivatives of an arbitrary functional with respect to these parameters are obtained, using the mixed approach in which both the direct and adjoint first-order solutions are used. The general results are particularized for the case of the complementary energy of a structure.

SHAPE OPTIMIZATION OF A 2D BODY SUBJECTED TO SEVERAL LOADING CONDITIONS

An algorithm for shape optimization based on simultaneous solution of the equations and inequalities arising from Kuhn-Tucker necessary conditions is presented. Regular triangular FE assembly is proposed. Element vertices are associated with design variables directly or through spline parameters defining the boundary of the optimized body. This way, during the iteration procedure, FE assembly is automatically remeshed together with the motion of the optimized boundary. Multiple loading conditions are represented in the problem as equality conditions in the form of a set of equilibrium equations for each loading condition separately. From the necessary condition equations an additional, important relation between cost function, Lagrange multipliers associated with inequality constraints and their limit values is derived. The algorithm combines standard professional FEM programs with an optimizer proposed in the paper which is illustrated with shape optimization of several 2D bodies. The proposed approach i...

Application of finite differences for solving the two-dimensional elasticity problem by means of the finite element method

Abstract Finite differences are applied to simplify the finite element method of solving the two-dimensional elasticity problem. Derivatives are replaced by difference quotients so that the only degrees of freedom are the values of the function at the nodes. Numerical examples are included.