Optimization of a hyper-elastic structure with multibody contact using continuum-based shape design sensitivity analysis

In this paper, a continuum-based shape design sensitivity formulation is presented for a hyper-elastic structure with multibody frictional contact. A nearly incompressible constraint is treated using the pressure projection method that projects a hydrostatic pressure into a lower order space to avoid a volumetric locking. The variational formulation for multibody frictional contact is developed using a penalty method that regularizes the solution of the variational inequality. The material derivative of continuum mechanics is utilized to develop the continuum-based shape design sensitivity analysis for the hyper-elastic constitutive relation and penalized contact formulation. The sensitivity equation is solved at each converged load step using the same tangent stiffness of response analysis due to the path dependency of the sensitivity of the frictional contact problem. A very accurate and efficient sensitivity results are shown through shape optimization of a windshield wiper. Received August 8, 1999     

Second-order shape design sensitivity using P-version finite element analysis

Thep-version finite element analysis (FEA) approach is attractive for design sensitivity analysis (DSA) and optimization due to its high accuracy of analysis results, even with coarse mesh; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape design changes during design iterations. A continuum second-order shape DSA formulation is derived and implemented usingp-version FEA. The second-order shape design sensitivity can be used for reliability based analysis and design optimization by incorporating it with the second-order reliability analysis method (SORM). Both the second-order shape DSA formulations with respect to the single and mixed shape design parameters are derived for elastic solids using the material derivative concept. Both the direct differentiation and hybrid methods are presented in this paper. A shape DSA is implemented by using an establishedp-version FEA code, STRESS CHECK. Two numerical examples, a connecting rod and bracket, are presented to demonstrate the feasibility and accuracy of the proposed seond-order shape DSA approach.     

Structural shape optimization of shells and folded plates using two-noded finite strips

This paper deals with structural shape optimization of shells and folded plates using two-noded Mindlin-Reissner C(0) finite strips. The whole shape optimization process is carried out by integrating finite strip analysis, cubic spline shape definition, automatic mesh generation, sensitivity analysis and mathematical programming methods in an efficient way. Both thickness and shape variables defining the cross-section of the structure are considered. The objective is to minimize the strain energy with a constraint that the total material volume of the structure remains constant. It is observed that minimization of strain energy leads to optimum structures in which the deflections and stress resultants in the members are considerably reduced. This is illustrated using several examples. The relative contributions of the bending, membrane and shear strain energies are also monitored during the whole optimization process. It is found that most optimal shell and folded plate structures are membrane dominant.     

Structural modelling and sensitivity analysis of shape optimization

It is presented in this paper that the structural modelling of shape optimization is composed of, in general cases, four distinct processes on geometry, design, analysis and perturbation models. The relationships between these models are discussed. An integrated modelling approach based on geometric shape parameterization and automatic mesh generation is proposed. In cooperation with this modelling approach, the semi-analytic sensitivity analysis has been effectively employed. These techniques join shape optimization with FEM and CAD packages and apply it versatilely to optimum designs of general structures. The implementation and applications of the integrated modelling approach and semi-analytic sensitivity analysis to shape optimization of structures with coupling of stress and temperature fields are illustrated.Presented at NATO ASI Optimization of Large Structural Systems, held in Berchtesgaden, Germany, Sept. 23 — Oct. 4, 1991     

Structural shape optimization using self-adjusted convex approximation

This study researches the applications of Self-Adjusted Convex Approximation (SACA) in structural shape optimization problems. The B-spline curve is adopted as the mathematical representation of the structural shapes. The SACA method is based on the CONvex LINearization (CONLIN) method and has better accuracy and convergent rate. Numerical examples are offered and the results show that the proposed method is effective in the structural shape design.     

Three-dimensional shape optimization of hip prostheses using a multicriteria formulation

A multicriteria optimization model is developed to obtain the optimal geometry of the femoral component of a hip prosthesis. The objective function minimizes both the relative tangential displacement and the contact normal stress. For cementless stems, these two factors are relevant for the prosthesis stability and therefore for the implant success. The three-dimensional optimization procedure developed allows us to characterize the stem shape that minimizes displacement and stress individually, or simultaneously using a multicriteria approach. Design variables characterize successive stem sections, and are subjected to linear geometric constraints to obtain clinically admissible geometries. Multiple loads are considered to incorporate several daily life activities. The system bone–stem is considered a structure in equilibrium with contact condition on the interface. Results show that thin stem tips minimize the interface stress while collared stems minimize displacement. The multicriteria formulation leads to balanced solutions.     

Shape design sensitivity analysis and optimization of general shape arches

A method is presented for the shape design sensitivity analysis as applied to general arches whose shapes cannot be mapped on one plane. The shape design sensitivity formulation with respect to the perturbation in the direction normal to the middle surface of the shallow arch curve is derived using the material derivative and the adjoint variable method with the system variational equation expressed in a Cartesian coordinate system. A general shape arch is subdivided into segments each of which can be considered as a shallow arch. On each subdivision of the arch, a Cartesian coordinate system is installed and the shape design sensitivity for the shallow arch is applied. For numerical implementation, the finite element method is adopted and each finite element can be considered as such a subdivision. Numerical examples for sensitivity analysis and optimization are presented to illustrate the finite element sensitivity method proposed.     

2D shape deformation using nonlinear least squares optimization

This paper presents a novel 2D shape deformation algorithm based on nonlinear least squares optimization. The algorithm aims to preserve two local shape properties: the Laplacian coordinates of the boundary curve and the local area of the shape interior, which are together represented in a non-quadratic energy function. An iterative Gauss–Newton method is used to minimize this nonlinear energy function. The result is an interactive shape deformation system that can achieve physically plausible results that are difficult to achieve with previous linear least squares methods. In addition to this algorithm that preserves local shape properties, we also introduce a scheme to preserve the global area of the shape, which is useful for deforming incompressible objects.     

A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh.An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes.The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation.Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

A consistent formulation for imposing packaging constraints in shape optimization using Vertex Morphing parametrization

This paper aims at imposing no-penetration condition over arbitrary surfaces which act as bounding surfaces, also known as packaging constraints, on the design surface of shape optimization problem. We use Vertex Morphing technique for the shape parametrization. Vertex Morphing is a consistent surface control approach for node-based shape optimization. The suitability of this technique has been assessed and demonstrated for a wide range of engineering applications without geometric shape constraints. In this contribution, a consistent formulation is presented for the implementation of numerous point-wise geometric constraints in four main steps. First, a potential contact between optimization surface points and the bounding surface is identified via the so-called gap function. Second, the shape gradients of objective functions and active constraints are mapped onto the Vertex Morphing’s control space, where the optimization problem is formulated. Third, the linear least squares method is used to project the steepest-descent search direction onto the subspace tangent to the mapped active constraints. Finally, the feasible design update is mapped onto the geometry space. To verify the perfect consistency between the geometry space (where the constraints are formulated) and the control space (where the optimization problem is solved) two applications of CFD shape optimization in the automotive industry are presented.     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Multi-objective shape design optimization of piezoelectric energy harvester using artificial immune system

Energy harvesting is about deriving energy from environment and converting into electricity. In this paper, optimal design of a cantilever piezoelectric energy harvester is presented with the aim to capture electrical power from a vibratory feeder in mining industry. Rayleigh–Ritz method is utilized for the modeling of the cantilever piezoelectric, taking into account possible variation in the width, nonequivalent layer lengths and thickness for unimorph and bimorph configurations. Innovatively, intelligent artificial immune system is utilized for multi-objective optimization of the shape parameters of the system. To verify the presented analytical shape optimization method, finite element analysis of the designed system is also presented, to investigate the output voltage and stress distribution along the piezoelectric layer. Moreover, the experimental setup is generated and verification tests are performed to derive frequency response diagram of the system. The obtained results are encouraging, indicating good agreement between experiments, FE analysis and theoretical results.     

Structural shape optimization integrated with CAD environment

The research work presented here is based on the concept of the integration of optimization techniques and numerical analysis with the finite element method (FEM) and computer-aided design (CAD). A microcomputer aided optimum design system, MCADS, has been developed for general structures. Certain techniques to be discussed in the paper, e.g. the semi-analytical method for design sensitivity analysis, optimization analysis modelling for shape design, application oriented user interfaces and the coupling of automated optimization and user intervention have rendered MCADS pratical and versatile in applications for engineering structures. The above techniques and an application are presented in this paper.     

Two dimensional shape optimization using partial control and finite element method for compressible flows

The objective of this study is to determine the two dimensional shape of a body located in a compressible viscous flow, where the applied fluid force is minimized. The formulation to obtain the optimal shape is based on an optimal control theory. An optimal state is defined as a state, in which the performance function defined as the integration of the square sum of the applied fluid forces is minimized due to a reduction in the applied fluid forces. Compressible Navier–Stokes equations are treated as constraint equations. In other words, the body is considered to have a shape that minimizes the fluid forces under the constraint of the Navier–Stokes equations. The gradient of the performance function is computed using the adjoint variables. A weighted gradient method is used as the minimization algorithm. The volume of the body is assumed to be the same as that of the initial body. In the case of the algorithm used in this study, both the creation of a structured mesh around the surface of the body and the smoothing procedure are employed for the computation of gradient. In this study, a remeshing technique based on the structured mesh around the body changing its configuration in the iteration cycle is employed. For the correction to keep the volume constant, the surface coordinates are moved along the radial direction. For the discretization of both the state and adjoint equations, the efficient bubble function interpolation presented previously by the authors [18] is employed. The algorithm, which is known as the partial control algorithm, is applied to the numerical procedure to determine the movement of the coordinates. In the case of the gradient method, in order to avoid the convergence of the final shape to the local minimum shape, the new algorithm, which is called the partial control algorithm, is presented in this study. In numerical studies, the shape determination of a body in a uniform flow field is carried out in 2D domains. The initial shape of the body is assumed to be an elliptical cylinder. The shape is modified by minimizing the applied fluid forces. Finally, the desired shape of a body, whose performance function is reduced and converged to a constant value, is obtained. By carrying out a procedure that involves the use of the partial control algorithm, the desired shape of a body, whose performance function is reduced further, is obtained. Stable shape determination of a body in a compressible viscous flow is carried out by using the presented method. It is indicated that the optimal shape can be obtained by using the partial control algorithm.     

Robust formulation for Reliability-based design optimization of structures

The reliability-based design optimization (RBDO) has been widely recognized as a powerful optimization tool under probabilistic constraints, through appropriate modeling of uncertainties. However, the drawback of RBDO is that it does not reflect the ability of the structure to comply with large data variations, unforeseen actions or deterioration mechanisms. On the other hand, the robust design optimization (RDO) reduces the variability of the structural performance, in addition to its mean level. However, RDO does not take direct advantage of the interaction between controllable (product design values) and noise variables (environmental random values), and the obtained results do not accurately indicate what parameter has the highest effect on the performance characteristics. The purpose of this paper is to propose a robust formulation for reliability-based design optimization (RRBDO) that combines the advantages of both optimization procedures and overcomes their weaknesses. The optimization model proposed overcomes the limitations of the existing models without compromising the reliability level, by considering a robust convex objective function and a performance variation constraint. The proposed formulation can consider the total cost of structures and can control structural parameter variations. It takes into account uncertainty and variability in the same mathematical formulation. A numerical solution procedure is also developed, for which results are analyzed and compared with RBDO for several examples of concrete and steel structures.