Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Modelling and process optimization for functionally graded materials

We optimize continuous quench process parameters to produce functionally graded aluminium alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard non‐linear programming algorithm. Ingredients of this algorithm include (1) the process parameters to be optimized, (2) a cost function: the weighted average of the precipitate number density distribution, (3) constraint functions to limit the temperature gradient (and hence distortion and residual stress) and exit temperature, and (4) their sensitivities with respect to the process parameters. The cost and constraint functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction‐rate theory to determine the precipitate particle sizes and their distributions. Both the temperature and the precipitate models are solved via the discontinuous Galerkin finite element method. The energy balance incorporates non‐linear boundary conditions and material properties. The temperature field is then used in the reaction rate model which has as many as 10⁵ degrees‐of‐freedom per finite element node. After computing the temperature and precipitate size distributions we must compute their sensitivities. This seemingly intractable computational task is resolved thanks to the discontinuous Galerkin finite element formulation and the direct differentiation sensitivity method. A three‐dimension example is provided to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Implementation of an exact finite reduction scheme for steady‐state reaction‐diffusion equations

In this paper we propose the numerical solution of a steady‐state reaction‐diffusion problem by means of application of a non‐local Lyapunov–Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non‐linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non‐linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple non‐linear one‐dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

Bilateral filtering for structural topology optimization

Filtering has been a major approach used in the homogenization‐based methods for structural topology optimization to suppress the checkerboard pattern and relieve the numerical instabilities. In this paper a bilateral filtering technique originally developed in image processing is presented as an efficient approach to regularizing the topology optimization problem. A non‐linear bilateral filtering process leads to a suitable problem regularization to eliminate the checkerboard instability, pronounced edge preserving smoothing characteristics to favour the 0–1 convergence of the mass distribution, and computational efficiency due to its single pass and non‐iterative nature. Thus, we show that the application of the bilateral filtering brings more desirable effects of checkerboard‐free, mesh independence, crisp boundary, computational efficiency and conceptual simplicity. The proposed bilateral technique has a close relationship with the conventional domain filtering and range filtering. The proposed method is implemented in the framework of a power‐law approach based on the optimality criteria and illustrated with 2D examples of minimum compliance design that has been extensively studied in the recent literature of topology optimization and its efficiency and accuracy are highlighted. Copyright © 2005 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

Regularization in model updating

This paper presents the theory of sensitivity‐based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters. Model updating, together with the additional regularization criterion, is an optimization with two objective functions, and must be linearized to obtain the solution. Structured solutions are obtained, based on the generalized singular value decomposition (GSVD), and specific features of the parameter and response paths as the regularization parameter varies are explored. The four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families. These concepts are demonstrated on a simulated discrete example and on an experimental case study. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.     

Design sensitivity analysis and optimization for minimizing springback of sheet‐formed part

The springback is a manufacturing defect in the stamping process and causes difficulty in product assembly. An impediment to the use of lighter‐weight, higher‐strength materials in manufacturing is relative lack of understanding about how these materials respond to complex forming processes. The springback can be reduced by using an optimized combination of die, punch, and blank holder shapes together with friction and blank‐holding force. An optimized process can be determined using a gradient‐based optimization to minimize the springback. For an effective optimization of the stamping process, development of an efficient design sensitivity analysis (DSA) for the springback with respect to these process parameters is crucial. A continuum‐based shape and configuration DSA method for the stamping process has been developed using a non‐linear shell model. The material derivative is used to develop the continuum‐based design sensitivity. The design sensitivity equation is solved without iteration at each converged load step in the finite deformation elastoplastic non‐linear analysis with frictional contact, which makes sensitivity calculation very efficient. Numerical implementation of the proposed shape and configuration DSA method is performed using the meshfree method. The accuracy and efficiency of the proposed method are illustrated by minimizing the springback in a benchmark S‐rail problem. Copyright © 2007 John Wiley & Sons, Ltd.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

A simple integration algorithm for a non‐associated anisotropic plasticity model for sheet metal forming

In this paper, an anisotropic material model based on a non‐associated flow rule and nonlinear mixed isotropic‐kinematic hardening is developed. The quadratic Hill48 yield criterion is considered in the non‐associated model for both yield function and plastic potential to account for anisotropic behavior. The developed model is integrated based on fully implicit backward Euler's method. The resulting problem is reduced to only two simple scalar equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. All numerical development was implemented into user‐defined material subroutine for the commercial finite element code ABAQUS/Standard. The performance of the present algorithm is demonstrated by numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary layer in shape optimization of convective fins using a meshfree approach

We show how shape optimization of fin arrays for increased heat flux through the base under area constraint leads to non‐existence of optimal solutions. An additional constraint in terms of the boundary layer eliminates the apparent paradox. We consider a variable heat transfer coefficient and we use a fixed‐point iteration scheme to solve the resulting non‐linear boundary value problem for the steady‐state heat operator with temperature, flux, and convection boundary conditions. We propose a simple yet effective algorithm for evaluating the boundary layer constraint and eliminating the constraint violation. There are large shape changes between the initial and final design but no remeshing is required because we use a meshfree method that is not sensitive to shape distortion of integration cells as long as they remain convex. The resulting optimal unit cell is repeated by periodicity to produce the optimal fin array. The obtained shapes display similarities to shapes seen in natural systems governed by diffusion/convection and conduction processes. A length‐scale for the unit cell is naturally introduced by the non‐overlap condition imposed on the thermal boundary layer in the cooling ambient fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimization‐based stability analysis of structures under unilateral constraints

This paper discusses an optimization‐based technique for determining the stability of a given equilibrium point of the unilaterally constrained structural system, which is subjected to the static load. We deal with the three problems in mechanics sharing the common mathematical properties: (i) structures containing no‐compression cables; (ii) frictionless contacts; and (iii) elastic–plastic trusses with non‐negative hardening. It is shown that the stability of a given equilibrium point of these structures can be determined by solving a maximization problem of a convex function over a convex set. On the basis of the difference of convex functions optimization, we propose an algorithm to solve the stability determination problem, at each iteration of which a second‐order cone programming problem is to be solved. The problems presented are solved for various structures to determine the stability of given equilibrium points. Copyright © 2008 John Wiley & Sons, Ltd.     

Checkerboard‐free topology optimization using non‐conforming finite elements

The objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non‐conforming four‐node finite elements are employed. Since the convergence of the non‐conforming finite element is independent of the Lamé parameters, the stiffness of the non‐conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard‐free property of the non‐conforming element in topology optimization problems and verify it with three typical optimization examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A constrained non‐linear system approach for the solution of an extended limit analysis problem

A number of recent papers (see, e.g. (Int. J. Mech. Sci. 2007; 49 :454–465; Eur. J. Mech. A/Solids 2008; 27 :859–881; Eng. Struct. 2008; 30 :664–674; Int. J. Mech. Sci. 2009; 51 :179–191)) have shown that classical limit analysis can be extended to incorporate such important features as geometric non‐linearity, softening and various so‐called ductility constraints. The generic formulation takes the form of a challenging (nonconvex and nonsmooth) optimization problem referred to in the mathematical programming literature as a mathematical program with equilibrium constraints (MPEC). Similar to a classical limit analysis, the aim is to compute in a single step a bound (upper bound, in the case of the extended problem) to the maximum load. The solution algorithm so far proposed to solve the MPEC is to convert it into an iterative non‐linear programming problem and attempts to process this using a standard non‐linear optimizer. Motivated by the fact that no method is guaranteed to solve such MPECs and by the need to avoid the use of an optimization approach, which is unfamiliar to most practising engineers, we propose, in the present paper, a novel numerical scheme to solve the MPEC as a constrained non‐linear system of equations. We illustrate the application of this approach using the simple class of elastoplastic softening skeletal structures for which certain ductility conditions are prescribed. Copyright © 2009 John Wiley & Sons, Ltd.     

Topology synthesis of large‐displacement compliant mechanisms

This paper describes the use of topology optimization as a synthesis tool for the design of large‐displacement compliant mechanisms. An objective function for the synthesis of large‐displacement mechanisms is proposed together with a formulation for synthesis of path‐generating compliant mechanisms. The responses of the compliant mechanisms are modelled using a total Lagrangian finite element formulation, the sensitivity analysis is performed using the adjoint method and the optimization problem is solved using the method of moving asymptotes. Procedures to circumvent some numerical problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Large‐deformation and friction analysis of non‐linear elastic cable networks by second‐order cone programming

A new formulation is presented for equilibrium shape analysis of cable networks considering geometrical and material non‐linearities. Friction between cables and joint devices is also considered. The second‐order cone programming (SOCP) problem which has the same solution as that of minimization of total potential energy is solved to obtain the equilibrium configuration. The optimality conditions are derived to verify that the solution satisfies equilibrium conditions and friction laws. Since no assumption on stress state is needed in the proposed method, no process of trial and error is involved. No effort is required to develop any analysis software because SOCP can be solved by using the well‐developed software based on the interior‐point method. Copyright © 2002 John Wiley & Sons, Ltd.     

Efficient non‐linear reanalysis of skeletal structures using combined approximations

Non‐linear reanalysis of large‐scale structures usually involves much computational effort, because the set of non‐linear equations must be solved repeatedly during the solution process. Various approximations that are often used for linear reanalysis are not sufficiently accurate for non‐linear problems. In this study, solution procedures based on the combined approximations approach are developed and compared in terms of efficiency and accuracy. Various path‐independent non‐linear analysis and reanalysis problems are considered, including material non‐linearity, geometric non‐linearity and buckling analysis. Numerical examples demonstrate the effectiveness of the procedures presented. It is shown that in various cases accurate results can be achieved efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

Improvement of semi‐analytical design sensitivities of non‐linear structures using equilibrium relations

The accuracy problem of the semi‐analytical method for shape design sensitivity analysis has been reported for linear and non‐linear structures. The source of error is the numerical differentiation of the element internal force vector, which is inherent to the semi‐analytical approach. Such errors occur for structures whose displacement field is characterized by large rigid body rotations of individual elements. This paper presents a method for the improvement of semi‐analytical sensitivities. The method is based on the element free body equilibrium conditions, and on the exact differentiation of the rigid body modes. The method is efficient, simple to code, and can be applied to linear and non‐linear structures. The numerical examples show that this approach eliminates the abnormal errors that occur in the conventional semi‐analytical method. Copyright © 2001 John Wiley & Sons, Ltd.     

On a wildland fire model with radiation

The 2‐D simplified wildland fire model presented here is based on conservation laws and takes into account radiation as the dominant thermal transfer mechanism, as well as convection, which represents the effect of the wind and the slope. The non‐dimensional equations are obtained using the Frank–Kamenestkii change of variables. The existence of weak solutions of the non‐linear reaction diffusion problem obtained is established as a particular case of more general existence results using a compactness method. The uniqueness of the weak solution is shown for a given initial data and fixed parameters of the equations. The approximate solution is obtained using a mixed finite element method. This preserves the continuity of the flux through the inter‐element boundaries and allows to represent high gradients in the solution. Semi‐discrete error estimation is presented. The convective term is solved by a splitting technique using Godunov's method. The numerical examples show the efficiency of the algorithm in capturing the fire front, even for critical values of the parameters. Copyright © 2002 John Wiley & Sons, Ltd.