AN INTERACTIVE OBJECT-ORIENTED TOOL FOR STRUCTURAL OPTIMIZATION

COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. O˜ nate and E. Dvorkin (Eds.) c CIMNE, Barcelona, Spain 1998
AN INTERACTIVE OBJECT-ORIENTED TOOL FOR STRUCTURAL OPTIMIZATION
Cl´ audio A. de Carvalho Silva and Marco L. Bittencourt
Universidade Estadual de Campinas
Faculdade de Engenharia Mecˆ anica, Departamento de Projeto Mecˆ anico P.O. Box 6051, Zip Code 13083-970, Campinas/SP, Brazil e-mail:cacs@fem.unicamp.br e-mail:mlb@fem.unicamp.br Key Words: Optimization, sensitivity analysis, object-oriented programming, C++, finite elements, Nurbs. Abstract. An interactive shape and parameter optimization tool was developed using continuum sensitivity analysis, mathematical programming and elastic structural analysis. Global efficiency of an optimization tool not only depends on the kind of algorithms used but also on the effectiveness of the data exchange among its several modules during the iterative process. Object-oriented programming features were applied in the development of an structural optimization enviroment aiming for efficiency. The evaluation of performance functionals and sensitivity analysis procedures were implemented in a class having the finite element model of the system to be optimized as a variable. Hence, the characteristics of the finite element model, its methods of solution and results are stored in memory, allowing them to be accessed without data file sharing costs. All these numerical tools were integrated by means of a NURBS-based graphical interface.
where a (u, v ) = B T (u) · E (v ) dV ; l (v ) = ∂ B t·v dA + B b·v dV ; T is the stress tensor; E is the strain tensor; t is the surface force ; b is the body force; V is the set of all kinematically possible solutions. Applying the Galerkin method an approximated solution is computed u(N ) in the N -dimensional subspace VN ⊂ V such that u ≈ u(N ) = (N ) (N ) N i=1 ui φi , VN = span [φ1 , . . . , φN ], where ui , i = 1, . . . , N are scalars. Observe that u= 2.2 u1
(1)
f , gi e hj are the structural performance functionals of the problem: f is the objective 2
Cl´ audio A. de Carvalho Silva and Marco L. Bittencourt
In applications, an optimization code works together with an analysis program. In structural engineering field, the most widely used technique is, undoubtedly, the finite element method, because of its accuracy and versatility. Therefore, it is worthwhile to examine the characteristics of an optimization problem whose analysis stage is done by finite elements. Consider the following model of structural optimizatiom problem,8 min f (p, u (p)) sujeita a ` gi (p, u (p)) ≤ 0 hj (p, u (p)) = 0 i = 1, 2, . . . , m j = i e hj are the constraints. The vector of nodal displacements u ∈ N satisfies the finite element equilibrium equation K (p) u = F (p), where p ∈ Ω ⊂ n is the ndimensional vector of design variables, K (p) is the stiffness matrix and F (p) is a vector of applied loads. That linear system represents the finite element solution of the variational problem of linear elasticity defined on the domain B ∈ 3 : Find the displacement field u ∈ V such that a (u, v ) = l (v ) ∀v ∈ V (2)
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Cl´ audio A. de Carvalho Silva and Marco L. Bittencourt
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INTRODUCTION
In spite of object-oriented programming (OOP) framework being well suited to deal with large software, there still are few implementations of this methodology in engineering. To some extent, it is due to the loss of efficiency the first object-oriented languages brought to implementations. Even using the present languages such as C++, which perform as well as traditional procedural languages, the careless application of OOP features can compromise the numerical efficiency of the code when compared to traditional practices. The aim of this work is to present the concepts used and the results obtained in the development of an structural optimization tool, which applies OOP features to reach high global efficiency while keeping the benefits of this programming technique. An interior point mathematical programming algorithm1–3 was considered. The finite element method is used for structural analysis and gradients of performance functionals gradients are computed by continuum-based design sensitivity analysis.4 The design geometry is described by NURBS (Non Uniform Ration B-Splines) curves due to their flexibility and simplicity on the description of boundaries.5, 6 The computational requirements of an efficient structural optimization enviroment are identified from the mathematical formulation of the optimization problem. One of the characteristics of this problem is the necessity of dealing with complex data structures used for finite element analysis and geometry description. In this way, improvements on the global efficiency can be reach with clever access, management and sharing of data among components of the environment. As OOP features very good characteristics of data modelling, it can be used to obtain better reliability and efficiency. In addition, combining a NURBS-based interface, interactive finite element, parametrization and performance functional definition can bring productivity to structural design. 2 2.1 MATHEMATICAL FORMULATION OF STRUCTURAL OPTIMIZATION Structural optimization model