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  • 1.
    Amouzgar, Kaveh
    et al.
    Product Development Department, School of Engineering, Jönköping University, Jönköping, Sweden; School of Engineering Science, University of Skövde, Skövde, Sweden.
    Strömberg, Niclas
    Örebro University, School of Science and Technology. Department of Mechanical Engineering.
    Radial Basis Functions as Surrogate Models with A Priori Bias in Comparison with a Posteriori Bias2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 4, p. 1453-1469Article in journal (Refereed)
    Abstract [en]

    In order to obtain a robust performance, the established approach when using radial basis function networks (RBF) as metamodels is to add a posteriori bias which is defined by extra orthogonality constraints. We mean that this is not needed, instead the bias can simply be set a priori by using the normal equation, i.e. the bias becomes the corresponding regression model. In this paper we demonstrate that the performance of our suggested approach with a priori bias is in general as good as, or even for many test examples better than, the performance of RBF with a posteriori bias. Using our approach, it is clear that the global response is modelled with the bias and that the details are captured with radial basis functions. The accuracy of the two approaches are investigated by using multiple test functions with different degrees of dimensionality. Furthermore, several modeling criteria, such as the type of radial basis functions used in the RBFs, dimension of the test functions, sampling techniques and size of samples, are considered to study their affect on the performance of the approaches. The power of RBF with a priori bias for surrogate based design optimization is also demonstrated by solving an established engineering benchmark of a welded beam and another benchmark for different sampling sets generated by successive screening, random, Latin hypercube and Hammersley sampling, respectively. The results obtained by evaluation of the performance metrics, the modeling criteria and the presented optimal solutions, demonstrate promising potentials of our RBF with a priori bias, in addition to the simplicity and straight-forward use of the approach.

  • 2.
    Dai, Xiaoxia
    et al.
    School of Computing Science, Zhejiang University City College, Hangzhou, People’s Republic of China.
    Zhang, Chengwei
    School of Computing Science, Zhejiang University City College, Hangzhou, People’s Republic of China.
    Zhang, Ye
    Örebro University, School of Science and Technology.
    Gulliksson, Mårten
    Örebro University, School of Science and Technology.
    Topology optimization of steady Navier-Stokes flow via a piecewise constant level set method2018In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 57, no 6, p. 2193-2203Article in journal (Refereed)
    Abstract [en]

    This paper presents a piecewise constant level set method for the topology optimization of steady Navier- Stokes flow. Combining piecewise constant level set functions and artificial friction force, the optimization problem is formulated and analyzed based on a design variable. The topology sensitivities are computed by the adjoint method based on Lagrangian multipliers. In the optimization procedure, the piecewise constant level set function is updated by a new descent method, without the needing to solve the Hamilton-Jacobi equation. To achieve optimization, the piecewise constant level set method does not track the boundaries between the different materials but instead through the regional division, which can easily create small holes without topological derivatives. Furthermore, we make some attempts to avoid updating the Lagrangian multipliers and to deal with the constraints easily. The algorithm is very simple to implement, and it is possible to obtain the optimal solution by iterating a few steps. Several numerical examples for both two- and three-dimensional problems are provided, to demonstrate the validity and efficiency of the proposed method.

  • 3.
    Gustafsson, Erik
    et al.
    SweCast AB, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Shape optimization of castings by using successive response surface methodology2008In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 35, no 1, p. 11-28Article in journal (Refereed)
    Abstract [en]

    In this paper, an optimization routine for a thermomechanical problem is presented. The optimization routine is based on the successive response surface methodology where the panning and zooming technique presented by Stander and Craig has been implemented and improved. The optimization routine has been applied to an optimization problem of a three-dimensional beam that undergoes a solidification process. The material in the beam is assumed to be low-alloyed gray iron. The thermomechanical solidification analysis is uncoupled where, first, a thermal analysis is performed to determine the thermal history. This thermal history is then used to calculate the residual stresses in the beam. The residual stresses are solved by using classical J(2)-plasticity with temperature-dependent material properties. The residual stresses from solidification are then carried on to the structural analysis where a mechanical load is applied. These are all linked together via scripts, and the commercial FE software Abaqus is used as the FE solver. The obtained maximum von Mises stress and mass information for every set of parameters are then exported to Matlab where general quadratic response surfaces are fitted by a least square method. Taken together, these response surfaces define a minimum of weight problem, which is solved by using sequential linear programming. To minimize the number of evaluations needed, the parameters are chosen to be D-optimally selected. The numerical results show that the residual stresses from solidification might influence the optimal shape significantly. The residual stress results have been compared with those obtained from casting simulation softwares, and the results are similar. The optimization has been compared with a commercial optimization software and shows very promising results.

  • 4.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method2010In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 42, no 1, p. 73-88Article in journal (Refereed)
    Abstract [en]

    In this work a hybrid method of a genetic algorithm  and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for four numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.

  • 5.
    Klarbring, Anders
    et al.
    Department of Management and Engineering, The Institute of Technology, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    A note on the min-max formulation of stiffness optimization including non-zero prescribed displacements2012In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 45, no 1, p. 147-149Article in journal (Refereed)
    Abstract [en]

    The present theoretical note shows how a naturalobjective function in stiffness optimization, including bothprescribed forces and non-zero prescribed displacements,is the equilibrium potential energy. It also shows how theresulting problem has a saddle point character that may beutilized when calculating sensitivities.

  • 6.
    Klarbring, Anders
    et al.
    Department of Management and Engineering, The Institute of Technology, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology optimization of hyperelastic bodies including non-zero prescribed displacements2013In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 47, no 1, p. 37-48Article in journal (Refereed)
    Abstract [en]

    Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.

  • 7.
    Strömberg, Niclas
    Örebro University, School of Science and Technology.
    Reliability-based design optimization using SORM and SQP2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 56, no 3, p. 631-645Article in journal (Refereed)
    Abstract [en]

    In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.

  • 8.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance–volume product2010In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 42, no 3, p. 341-350Article in journal (Refereed)
    Abstract [en]

    In this paper the concept of extended optimality, or hyperoptimality, is adopted. By following this idea, a new compliance-volume product is suggested as objective. The volume appearing in the product is also raised to the power of a new design parameter which can be set to different values. In such manner design concepts with different volume fractions can be generated by using the approach of extended optimality. Both manufacturing constraints and unilateral contact constraints are included in the proposed method. The manufacturing constraints are implemented by adjusting the move limits such that the draw directions are satisfied. Both one draw direction as well as split draw constraints are considered. The contact conditions are modeled by the augmented Lagrangian approach such that the Jacobian in the Newton algorithm as well as in the adjoint equation becomes symmetric. The design parametrization is done by the SIMP model and Sigmund's filter is utilized when the sensitivities are calculated. The proposed method is very robust and efficient. This is demonstrated by solving problems in both 2D and 3D. The numerical results are also compared to solutions obtained by performing compliance optimization with a constraint on the volume fraction.

  • 9.
    Strömberg, Niclas
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Klarbring, Anders
    Division of Mechanics, Linköping University, Linköping, Sweden.
    Topology optimization of structures in unilateral contact2010In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 41, no 1, p. 57-64Article in journal (Refereed)
    Abstract [en]

    In this paper a general framework for topology optimization of structures in unilateral contact is developed. A linear elastic structure that is unilaterally constrained by rigid supports is considered. The supports are modeled by Signorini's contact conditions which in turn are treated by the augmented Lagrangian approach as well as by a smooth approximation. The latter approximation must not be confused with the well-known penalty approach. The state of the system, which is defined by the equilibrium equation and the two different contact formulations, is solved by a Newton method. The design parametrization is obtained by using the SIMP-model. The minimization of compliance for a limited value of volume is considered. The optimization problems are solved by SLP. This is done by using a nested approach where the state equations are linearized and sensitivities are calculated by the adjoint method. In order to avoid mesh-dependency the sensitivities are filtered by Sigmund's filter. The final LP-problem is solved by an interior point method that is available in Matlab. The implementation is done for a general design domain in 2D as well as in 3D by using fully integrated isoparametric elements. The implementation seems to be very efficient and robust.

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