| 1. |
- Amouzgar, Kaveh, 1980-, et al.
(författare)
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An approach towards generating surrogate models by using RBFN with a priori bias
- 2014
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Ingår i: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 2014, Vol. 2B. - New York, USA : ASME Press. - 9780791846322
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Konferensbidrag (refereegranskat)abstract
- In this paper, an approach to generate surrogate modelsconstructed by radial basis function networks (RBFN) with a prioribias is presented. RBFN as a weighted combination of radialbasis functions only, might become singular and no interpolationis found. The standard approach to avoid this is to add a polynomialbias, where the bias is defined by imposing orthogonalityconditions between the weights of the radial basis functionsand the polynomial basis functions. Here, in the proposed a prioriapproach, the regression coefficients of the polynomial biasare simply calculated by using the normal equation without anyneed of the extra orthogonality prerequisite. In addition to thesimplicity of this approach, the method has also proven to predictthe actual functions more accurately compared to the RBFNwith a posteriori bias. Several test functions, including Rosenbrock,Branin-Hoo, Goldstein-Price functions and two mathematicalfunctions (one large scale), are used to evaluate the performanceof the proposed method by conducting a comparisonstudy and error analysis between the RBFN with a priori and aposteriori known biases. Furthermore, the aforementioned approachesare applied to an engineering design problem, that ismodeling of the material properties of a three phase sphericalgraphite iron (SGI) . The corresponding surrogate models arepresented and compared
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| 2. |
- Amouzgar, Kaveh, 1980-, et al.
(författare)
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Multi-objective optimization of a disc brake system by using SPEA2 and RBFN
- 2013
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Ingår i: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. - New York : ASME Press. - 9780791855898
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)abstract
- Many engineering design optimization problems involve multiple conflicting objectives, which today often are obtained by computational expensive finite element simulations. Evolutionary multi-objective optimization (EMO) methods based on surrogate modeling is one approach of solving this class of problems. In this paper, multi-objective optimization of a disc brake system to a heavy truck by using EMO and radial basis function networks (RBFN) is presented. Three conflicting objectives are considered. These are: 1) minimizing the maximum temperature of the disc brake, 2) maximizing the brake energy of the system and 3) minimizing the mass of the back plate of the brake pad. An iterative Latin hypercube sampling method is used to construct the design of experiments (DoE) for the design variables. Next, thermo-mechanical finite element analysis of the disc brake, including frictional heating between the pad and the disc, is performed in order to determine the values of the first two objectives for the DoE. Surrogate models for the maximum temperature and the brake energy are created using RBFN with polynomial biases. Different radial basis functions are compared using statistical errors and cross validation errors (PRESS) to evaluate the accuracy of the surrogate models and to select the most accurate radial basis function. The multi-objective optimization problem is then solved by employing EMO using the strength Pareto evolutionary algorithm (SPEA2). Finally, the Pareto fronts generated by the proposed methodology are presented and discussed.
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| 3. |
- Amouzgar, Kaveh, 1980-, et al.
(författare)
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Radial basis functions as surrogate models with a priori bias in comparison with a posteriori bias
- 2017
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Ingår i: Structural and multidisciplinary optimization (Print). - : Springer Science and Business Media LLC. - 1615-147X .- 1615-1488. ; 55:4, s. 1453-1469
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 4. |
- Christensen, P. W., et al.
(författare)
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Formulation and comparison of algorithms for frictional contact problems
- 1998
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Ingår i: International Journal for Numerical Methods in Engineering. - : John Wiley & Sons. - 0029-5981 .- 1097-0207. ; 42:1, s. 145-173
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents two algorithms for solving the discrete, quasi-static, small-displacement, linear elastic, contact problem with Coulomb friction. The algorithms are adoptions of a Newton method for solving B-differentiable equations and an interior point method for solving smooth, constrained equations. For the application of the former method, the contact problem is formulated as a system of B-differentiable equations involving the projection operator onto sets with simple structure; for the application of the latter method, the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non-negativity of variables treated as constraints. The two algorithms are numerically tested for two-dimensional problems containing up to 100 contact nodes and up to 100 time increments. Results show that at the present stage of development, the Newton method is superior both in robustness and speed. Additional comparison is made with a commercial finite element code.
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| 5. |
- Gustafsson, Erik, et al.
(författare)
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Residual stresses in a stress lattice : experiments and finite element simulations
- 2009
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Ingår i: Journal of Materials Processing Technology. - : Elsevier. - 0924-0136 .- 1873-4774. ; 209:9, s. 4320-4328
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Tidskriftsartikel (refereegranskat)abstract
- In this work residual stresses in a stress lattice are studied. The residual stresses are both measured and simulated. The stress lattice is casted of low alloyed grey cast iron. In fact, nine similar lattices are casted and measured. The geometry of the lattice consists of three sections in parallel. The diameter of the two outer sections are thinner than the section in the middle. When the stress lattice cools down, this difference in geometry yields that the outer sections start to solidify and contract before the section in the middle. Finally, an equilibrium state, with tensile stresses in the middle and compressive stresses in the outer sections, is reached. The thermo-mechanical simulation of the experiments is performed by using Abaqus. The thermo-mechanical solidification is assumed to be uncoupled. First a thermal analysis, where the lattice is cooled down to room temperature, is performed. Latent heat is included in the analysis by letting the fraction of solid be a linear function of the temperature in the mushy zone. After the thermal analysis a quasi-static mechanical analysis is performed where the temperature history is considered to be the external force. A rate independent J2-plasticity model with isotropic hardening is considered, where the material data depend on the temperature. Tensile tests are performed at room temperature, 200°C, 400°C, 600°C and 800°C in order to evaluate the Young´s modulus, the yield strength and the hardening accurate. In addition, the thermal expansion coefficient is evaluated for temperatures between room temperature and 1000°C. The state of residual stresses is measured by cutting the mid section or the outer section. The corresponding elastic spring-back reveals the state of residual stresses. The measured stresses are compared to the numerical simulations. The simulations show good agreement with the results from the experiments.
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| 6. |
- Gustafsson, Erik, et al.
(författare)
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Shape Optimization of Castings by using Successive Response Surface Methodology
- 2008
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Ingår i: Structural and multidisciplinary optimization (Print). - : Springer Science and Business Media LLC. - 1615-147X .- 1615-1488. ; 35:1, s. 11-28
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
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| 8. |
- Gustafsson, Erik, et al.
(författare)
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Structural optimization of castings by using ABAQUS and Matlab
- 2005
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Ingår i: Competent Design by Castings. - Espoo, Finland : VTT. - 9513862984 - 9513862992 ; , s. 39-52
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Konferensbidrag (refereegranskat)abstract
- In this work a general method for structural optimization of nonlinear structures is implemented using FE-analysis. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled from a temperature above liquidus temperature down to room temperature. The thermomechanical analysis is uncoupled where the temperature distribution within the casting as a function of time is determined first and is later on used for residual stress calculations. These residual stresses are then included when the mechanical load is applied to the structure and the problem of minimum of weight is formulated. The structure shown in this paper is an example of a two dimensional geometry. The shape of the structures will of course affect the residual stress distribution during the optimization. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming.
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| 9. |
- Gustafsson, Erik, et al.
(författare)
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Structural optimization of castings by using Abaqus and Matlab
- 2005
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Ingår i: Proceedings of the Abaqus World Users' Conference.
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Konferensbidrag (refereegranskat)abstract
- In this work a general method for structural optimization of nonlinear structures is implemented using ABAQUS and Matlab. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. For instance plasticity problems, thermomechanical problems and contact problems can be optimized using this strategy. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled down from a temperature above liquidus temperature down to room temperature. These residual stresses are then included when the problem of minimum of weight is formulated. The shape of the structure will of course affect the residual stress distribution during the optimization and the optimal shape will be different from the one obtained when residual stresses are not included in the analysis. The method is implemented by using a Python script and m-files. In such way a parameterized model can easily be treated in ABAQUS and Matlab during the optimization process. The parameterized geometry, loads, boundary conditions and mesh are first generated by the ABAQUS/CAE module. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming where the linear part is solved using Matlab.
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| 10. |
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