Second-order shape sensitivity analysis for nonlinear problems

First- and second-order shape sensitivity analyses in a fully nonlinear framework are presented in this paper. Using the fixed domain technique and the adjoint approach, integral expressions over the domain are obtained. The Guillaume-Masmoudi lemma allows these expressions to be rewritten as integrals over the domain boundary. The formalism is then applied to the steady creep of a bar in torsion, as an example of power-law nonlinearities that occur not only in creep problems but also in viscoplastic fluid flow. Finally, a problem with known analytical solution is presented in order to show the equivalence between exact differentiation and the shape sensitivity approach.     

Shape sensitivity analysis using a fixed basis function finite element approach

An approach is presented for the determination of solution sensitivity to changes in problem domain or shape. A finite element displacement formulation is adopted and the point of view is taken that the finite element basis functions and grid are fixed during the sensitivity analysis; therefore, the method is referred to as a “fixed basis function” finite element shape sensitivity analysis. This approach avoids the requirement of explicit or approximate differentiation of finite element matrices and vectors and the difficulty or errors resulting from such calculations. Effectively, the sensitivity to boundary shape change is determined exactly; thus, the accuracy of the solution sensitivity is dictated only by the finite element mesh used. The evaluation of sensitivity matrices and force vectors requires only modest calculations beyond those of the reference problem finite element analysis; that is, certain boundary integrals and reaction forces on the reference location of the moving boundary are required. In addition, the formulation provides the unique family of element domain changes which completely eliminates the inclusion of grid sensitivity from the shape sensitivity calculation. The work is illustrated for some one-dimensional beam problems and is outlined for a two-dimensional C⁰ problem; the extension to three-dimensional problems is straight-forward. Received December 5, 1999?Revised mansucript received July 6, 2000     

A Partitioned finite element method for dynamical systems

A new analytical method is described for deriving the equations of motion of dynamical systems. The concept is to consider the displacements of the domain to be composed of rigid and elastic components. In contrast to other reduction methods, the domain modeled by finite number of degrees of freedom is discretized into two distinctive types of subdomains. Rigid and elastic subdomains are generated by consistent lumping of the domain properties under unique kinematic constraint relations. Equations of motion of the disjoint subdomains are derived by Lagrange's equations, in conjunction with the shape function matrix represented in partitioned form. This allows reduced sizes of matrices and avoids their possible singularities. Based on the invariance of energies under a compatible partitioned procedure, a simple analytical method is introduced for building the equations of motion of the whole domain from those of the subdomains. The dynamic analysis of a two-node domain with application to a blade-shaft combination is presented to illustrate the method.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

Design optimization of laminar flow machine rotors based on the topological derivative concept

Flow machines are very important to industry, being widely used on various processes. Thus, performance improvements are relevant and can be achieved by using topology optimization methods. In particular, this work aims to develop a topological derivative formulation to design radial flow machine rotors by considering laminar flow. Based on the concept of traditional topology optimization approaches, in the adopted topological derivative formulation, solid or fluid material is distributed at each point of the domain. This is achieved by combining Navier–Stokes equations on a rotary referential with Darcy’s law equations. This strategy allows for working in a fixed computational domain, which leads to a topology design algorithm of remarkably simple computational implementation. In the optimization problem formulation, a multi-objective function is defined, aiming to minimize the energy dissipation, vorticity and power considering a volume constraint. The constrained optimization problem is rewritten in the form of an unconstrained optimization problem by using the Augmented Lagrangian formalism. The resulting multi-objective shape functional is then minimized with help of the topological derivative concept. In the context of this article, the topological derivative represents the exact sensitivity with respect to the nucleation of an inclusion within the design domain and the obtained analytical (closed) formula can be evaluated through a simple post processing of the solutions to the direct and adjoints problems. Both mentioned features allow for obtaining the optimized designs in few iterations by using a minimal number of user defined algorithm parameters. All equations and the derived continuous adjoint equations are solved through finite element method. As a result, two-dimensional designs of flow machine rotors are obtained by using this methodology. Their performance is analyzed by evaluating velocity and pressure distributions inside rotor.     

Analytical expressions of sensitivities for shape variables in linear bending systems

Sensitivity analysis is a very interesting field in structural engineering because of its variety of uses. But the computational effort to obtain the analytical values of such sensitivities is a tough task that has been generally avoided when considering flexural systems. Instead some numerical approaches have been used to solve the problem. However, carrying out the sensitivity analysis by this method leads to considerable errors, especially with shape variables as many authors have pointed out. In this paper analytical expressions of sensitivities analysis with respect to shape variables are carried out for bending systems in linear theory. The development presented in this paper starts evaluating the sensitivity analysis of the nodal movements performing the loading vector and stiffness matrix sensitivity analysis. Then this research evaluates the sensitivity analysis of the maximum normal stresses. Finally, some structural examples where the previous analytical sensitivities are evaluated are exposed relating the results versus the corresponding results obtained by finite difference methods and some conclusions are drawn from the work presented.     

E(FG)<Superscript>2</Superscript>: a new fixed-grid shape optimization method based on the element-free galerkin mesh-free analysis

We propose a shape optimization method over a fixed grid. Nodes at the intersection with the fixed grid lines track the domain’s boundary. These “floating” boundary nodes are the only ones that can move/appear/disappear in the optimization process. The element-free Galerkin (EFG) method, used for the analysis problem, provides a simple way to create these nodes. The fixed grid (FG) defines integration cells for EFG method. We project the physical domain onto the FG and numerical integration is performed over partially cut cells. The integration procedure converges quadratically. The performance of the method is shown with examples from shape optimization of thermal systems involving large shape changes between iterations. The method is applicable, without change, to shape optimization problems in elasticity, etc. and appears to eliminate non-differentiability of the objective noticed in finite element method (FEM)-based fictitious domain shape optimization methods. We give arguments to support this statement. A mathematical proof is needed.     

High-fidelity aerostructural optimization with integrated geometry parameterization and mesh movement

This paper extends an integrated geometry parameterization and mesh movement strategy for aerodynamic shape optimization to high-fidelity aerostructural optimization based on steady analysis. This approach provides an analytical geometry representation while enabling efficient mesh movement even for very large shape changes, thus facilitating efficient and robust aerostructural optimization. The geometry parameterization methodology uses B-spline surface patches to describe the undeflected design and flying shapes with a compact yet flexible set of parameters. The geometries represented are therefore independent of the mesh used for the flow analysis, which is an important advantage to this approach. The geometry parameterization is integrated with an efficient and robust grid movement algorithm which operates on a set of B-spline volumes that parameterize and control the flow grid. A simple technique is introduced to translate the shape changes described by the geometry parameterization to the internal structure. A three-field formulation of the discrete aerostructural residual is adopted, coupling the mesh movement equations with the discretized three-dimensional inviscid flow equations, as well as a linear structural analysis. Gradients needed for optimization are computed with a three-field coupled adjoint approach. Capabilities of the framework are demonstrated via a number of applications involving substantial geometric changes.     

A Free Form Feature Taxonomy

In this paper the notion of free form feature for aesthetic design is presented. The design of industrial products constituted by free form surfaces is done by using CAD systems representing curves and surfaces by means of NURBS functions, which are usually defined by low level entities that are not intuitive and require some knowledge of the mathematical language. Similarly to the feature-based approach adopted by CAD systems for classical mechanical design, a set of high level modelling entities which provides commonly performed shape modifications has been identified. Particularly, the paper suggests a classification of the so-called detail features for an aesthetic and/or functional characterization of predefined free form surfaces. Feature types are formally described by means of an analytical definition of the surface modification through deformation and elimination laws. A topological classification is then given according to the application domain of such laws. A further sub-classification of morphological types is then suggested according to geometric properties of weak convexity and concavity for the resulting modified shape, leading to a taxonomy of simple free form features meaningful for aesthetic design.     

Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.     

Shape sensitivity analysis with design-dependent loadings—equivalence between continuum and discrete derivatives

The purpose of this paper is twofold: (1) showing equivalence between continuum and discrete formulations in sensitivity analysis when a linear velocity field is used and (2) presenting shape sensitivity formulations for design-dependent loadings. The equations for structural analysis are often composed of the stiffness part and the applied loading part. The shape sensitivity formulations for the stiffness part were well-developed in the literature, but not for the loading part, especially for body forces and surface tractions. The applied loads are often assumed to be conservative or design-independent. In shape design problems, however, the applied loads are often functions of design variables. In this paper, shape sensitivity formulations are presented when the body forces and surface tractions depend on shape design variables. Especially, the continuum–discrete (C–D) and discrete–discrete (D–D) approaches are compared in detail. It is shown that the two methods are theoretically and numerically equivalent when the same discretization, numerical integration, and linear design velocity fields are used. The accuracy of sensitivity calculation is demonstrated using a cantilevered beam under uniform pressure and an arch dam crown cantilever under gravity and hydrostatic loading at the upstream face of the structure. It is shown that the sensitivity results are consistent with finite difference results, but different from the analytical sensitivity due to discretization and approximation errors of numerical analysis.     

Shape optimal design using B-splines

Shape optimal design of an elastic structure is formulated using a design element technique. It is shown that Bezier and B-spline curves, typical of the CAD philosophy, are well suited to the definition of design elements. Complex geometries can be described in a very compact way by a small set of design variables and a few design elements. Because of the B-splines flexibility, it is no longer necessary to piece design elements together in order to agree with the shape complexity, nor to restrict the shape variations. Moreover, the additional optimization constraints that are most often needed to avoid unrealistic designs when the shape variables are the nodal coordinates of a finite element mesh, are automatically taken into account in the new formulation. An analytical derivation of the sensitivity analysis will be established, giving rise to numerical efficiency. It will be seen that the resulting optimization problem does not involve highly nonlinear functions with respect to the shape variables, so that simple mathematical programming algorithms can be applied to solve it. Some numerical examples are offered to demonstrate the power and generality of the new approach presented in this paper.     

Topological sensitivity analysis

The so-called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems. This derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain. However, the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study (because they have not the same topology). Therefore, some specific mathematical framework should be developed in order to obtain the derivatives. This work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts. The main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative. This idea leads to a more simple and constructive formulation than the ones found in the literature. Further, to point out the straightforward use of the proposed methodology, it is applied for solving some design problems in steady-state heat conduction.     

Shape sensitivities in the reliability analysis of nonlinear frame structures

A unified and comprehensive treatment of shape sensitivity that includes variations in the nodal coordinates, member cross-section properties, and global shape parameters of inelastic frame structures is presented. A novelty is the consideration of geometric uncertainty in both the displacement- and force-based finite element formulations of nonlinear beam-column behavior. The shape sensitivity equations enable a comprehensive investigation of the relative influence of uncertain geometrical imperfections on structural reliability assessments. For this purpose, finite element reliability analyses are employed with sophisticated structural models, from which importance measures are available. The unified approach presented herein is based on the direct differentiation method and includes variations in the equilibrium and compatibility relationships of frame finite elements, as well as the member cross-section geometry, in order to obtain complete shape sensitivity equations. The analytical shape sensitivity equations are implemented in the OpenSees software framework. Numerical examples involving a steel structure and a reinforced concrete structure confirm that geometrical imperfections may have a significant impact on structural reliability assessments.     

Numerical method for shape optimization using meshfree method

A numerical method for continuum-based shape design sensitivity analysis and optimization using the meshfree method is proposed. The reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method. Special features of the meshfree method from a sensitivity analysis viewpoint are discussed, including the treatment of essential boundary conditions, and the dependence of the shape function on the design variation. It is shown that the mesh distortion that exists in the finite element-based design approach is effectively resolved for large shape changing design problems through 2-D and 3-D numerical examples. The number of design iterations is reduced because of the accurate sensitivity information.     

Vibration sensitivity analysis of the ‘Butterfly-gyro’ structure

The ‘Butterfly-gyro’ is simple to manufacture with single sided electrostatic excitation and capacitive detection, and it is considered as one kind of the microgyroscope with high sensitivity due to its unique structure. This paper provides the sensitivity analytical model by solving the dynamic equations of motion and the design guidelines for microgyroscope with high sensitivity. Using Coriolis Effect and Newton’s second law, the dynamic equations are built. The sensitivity analytical model, including the denotations of Q factors and the resonant frequencies, is built. The approximate analytical expressions of Q factors and the resonant frequencies are derived by rational assumptions. Based on the sensitivity analytical model, the parametric analysis is carried out, and the design guidelines of high sensitivity are also deduced. Finally, Q factor, frequency split and other factors influencing the sensitivity are discussed in details to enhance its sensitivity. Results presented are valuable in the design and parameters optimization of the microgyroscope with high sensitivity.     

On shape sensitivity approaches in the numerical analysis of structures

The semi-analytical, analytical and direct methods for numerical structural shape sensitivity analysis are discussed for a beam model and the general three-dimensional case. While the two first methods are applied directly to the finite element model of a structure, the direct approach follows from a continuous formulation and only the final results can be discretized.