Rogério Leal

Boundary elements in shape optimal design of structural components

The shape optimal design of shafts and two-dimensional elastic structural components is formulated using boundary elements. The design objective is to maximize torsional rigidity of the shaft or to minimize compliance of the structure, subject to an area constrain Also a model based on minimum area and stress constraints is developed, where the real and adjoint structures are identical, but with different loading conditions. All degrees of freedom of the models are at the boundary and there is no need for calculating displacements and stresses in the domain. Formulations based on constant, linear and quadratic boundary elements are developed. A method for calculating accurately the stresses at the boundary is presented, which improves considerably the design sensitivity information. It is developed a technique for an automatic mesh refinement of boundary element models. The corresponding nonlinear programming problems are solved by Pshenichny’s linearization method. The models are applied to shape optimal design of several shafts and elastic structural components. The advantages and disadvantages of the boundary element method over the finite element technique for shape optimal design of structures are discussed with reference to applications. A literature survey of the development of the boundary element method for shape optimal design is presented.

Introduction to Finite Element Method

As discussed in Chap. 1, mechanic problems are governed by a set of partial differential equations that are valid in a certain domain and they needed to be solved for evaluating the stress condition of mechanical components. Although analytic methods can be employed to solve linear problems involving partial differential equations, its use to analyze complex structures may be a difficult or, even, an impossible task. Thus, in this chapter, Hamilton’s principle, which one of the most powerful energy principle, is introduced for the FEM formulation of problems of mechanics of solids and structures. The approach adopted in this chapter is to directly work out the dynamic system equations, after which the static dynamic equations can be easily obtained by simply dropping out the dynamic terms

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.

Adaptive boundary element method for bidimensional elasticity

Abstract An automatic technique for mesh refinement which improves the discretization and the accuracy of the boundary stress and strains is presented. These results are crucial for the shape optimal design of elasticity structures, in order to improve the design sensitivity analysis and to reduce the computational effort needed to find the optimal shape. The mesh refinement technique developed is based on a global and local equilibrium criterion.

Generalized Timoshenko modelling of composite beam structures: sensitivity analysis and optimal design

This article describes a new approach to design the cross-section layer orientations of composite laminated beam structures. The beams are modelled with realistic cross-sectional geometry and material properties instead of a simplified model. The VABS (the variational asymptotic beam section analysis) methodology is used to compute the cross-sectional model for a generalized Timoshenko model, which was embedded in the finite element solver FEAP. Optimal design is performed with respect to the layers’ orientation. The design sensitivity analysis is analytically formulated and implemented. The direct differentiation method is used to evaluate the response sensitivities with respect to the design variables. Thus, the design sensitivities of the Timoshenko stiffness computed by VABS methodology are imbedded into the modified VABS program and linked to the beam finite element solver. The modified method of feasible directions and sequential quadratic programming algorithms are used to seek the optimal continuous solution of a set of numerical examples. The buckling load associated with the twist–bend instability of cantilever composite beams, which may have several cross-section geometries, is improved in the optimization procedure.

Engineering Computation of Structures: The Finite Element Method

This book presents theories and the main useful techniques of the Finite Element Method (FEM), with an introduction to FEM and many case studies of its use in engineering practice. It supports engineers and students to solve primarily linear problems in mechanical engineering, with a main focus on static and dynamic structural problems. Readers of this text are encouraged to discover the proper relationship between theory and practice, within the finite element method:Practice without theory is blind, but theory without practice is sterile. Beginning with elasticity basic concepts and the classical theories of stressed materials, the work goes on to apply the relationship between forces, displacements, stresses and strains on the process of modeling, simulating and designing engineered technical systems. Chapters discuss the finite element equations for static, eigenvalue analysis, as well as transient analyses. Students and practitioners using commercial FEM software will find this book very helpful. It uses straightforward examples to demonstrate a complete and detailed finite element procedure, emphasizing the differences between exact and numerical procedures.

Actuator effect of a piezoelectric anisotropic plate model

This paper addresses the actuator effect of a piezoelectric anisotropic plate model, depending on the location of the applied electric potentials, and for different clamped boundary conditions. It corresponds to integer optimization problems, whose objective functions involve the displacement of the plate. We adopt the two-dimensional piezoelectric anisotropic nonhomogeneous plate model derived in Figueiredo and Leal [1]. This model is first discretized by the finite element method. Then, we describe the associated integer optimization problems, which aim to find the maximum mechanical displacement of the plate, as a function of the location of the applied electric potentials. In this sense, we also introduce a related multi-objective optimization problem, which is solved with genetic algorithms. Several numerical examples are reported. For all the tests, the stiffness matrices and force vectors have been evaluated with the subroutines planre and platre, of the CALFEM toolbox of MATLAB [2], and the genetic algorithms have been implemented in C ++ .

Mixed Elements in Shape Optimal Design of Structures Based on Global Criteria

In this paper mixed elements are applied to the optimal shape design of two dimensional elastic structures. The theory of shape sensitivity analysis of structures with differentiable objective functions is developed based on mixed elements, and using the Lagrangian approach and the material derivative concept. The mixed finite element model is based on an eight node mixed isoparametric quadratic element, whose degrees of freedom are two displacements and three stresses, per node. The corresponding nonlinear programming problem is solved using the method of sequential convex programming and the modified method of feasible directions, available in the commercial programme ADS (Automated Design Synthesis). The formulation developed is applied to the optimal shape design of two dimensional elastic structures, selecting the compliance as the objective function to be minimized and the initial volume as constraint. The advantages and disadvantages of the mixed elements are discussed with reference to applications.

Finite Element Method for Beams

A beam is a structural member whose geometry is very similar to the geometry of a bar. It is also geometrically a bar of an arbitrary cross-section, by bar it is meant that one of the dimensions is considerably larger than the other two, whose primary function is to support transverse loading. The main difference between the beam and the truss is the type of load that they support. In fact, beams are the most common type of structural component, especially in civil and mechanical engineering. A beam resists to transverse loads mainly through a bending action and, the bending is responsible for compressive longitudinal stresses in one side of the beam and tensile stress on the other beam side. These two regions are separated by the neutral axis in which the stress is zero. The combination of tensile and compressive stresses produces an internal bending moment. Finite element equations for beam-like structures are developed in this chapter.

Advanced FEM Modelling

This chapter presents a discussion on some modelling techniques for the stress analyses of solids and structures. Mesh symmetry, rigid elements and constraint equations, mesh compatibility, modelling of offsets, supports and connections between elements with different mathematical bases are all covered. Advanced modelling of laminated composite materials are also presented.