SQG-Differential Evolution for difficult optimization problems under a tight function evaluation budget

In the context of industrial engineering, it is important to integrate efficient computational optimization methods in the product development process. Some of the most challenging simulation-based engineering design optimization problems are characterized by: a large number of design variables, the absence of analytical gradients, highly non-linear objectives and a limited function evaluation budget. Although a huge variety of different optimization algorithms is available, the development and selection of efficient algorithms for problems with these industrial relevant characteristics, remains a challenge. In this communication, a hybrid variant of Differential Evolution (DE) is introduced which combines aspects of Stochastic Quasi-Gradient (SQG) methods within the framework of DE, in order to improve optimization efficiency on problems with the previously mentioned characteristics. The performance of the resulting derivative-free algorithm is compared with other state-of-the-art DE variants on 25 commonly used benchmark functions, under tight function evaluation budget constraints of 1000 evaluations. The experimental results indicate that the new algorithm performs excellent on the 'difficult' (high dimensional, multi-modal, inseparable) test functions. The operations used in the proposed mutation scheme, are computationally inexpensive, and can be easily implemented in existing differential evolution variants or other population-based optimization algorithms by a few lines of program code as an non-invasive optional setting. Besides the applicability of the presented algorithm by itself, the described concepts can serve as a useful and interesting addition to the algorithmic operators in the frameworks of heuristics and evolutionary optimization and computing

Global Optimization Algorithms in Multidisciplinary DesignOptimization

While Multidisciplinay Design Optimization (MDO) literature focuses mainly on the development of different formulations, through the manipulation of design variables, less attention is generally devoted to the combination of specific MDO formulations with existing nonlinear optimization algorithms. In this paper, the focus is on the application of a Global Optimization (GO) algorithm to an MDO problem. We first introduce and describe some MDO approaches from the literature. Then, we consider our MDO formulation where we deal with the GO box-constrained problem min_{a R We assume that the solution of the latter problem requires the use of a derivative-free methods since the derivatives of f(x) are unavailable and/or the function must be treated as a `black-box'. Within this framework we study some globally convergent modifications of the evolutionary Particle Swarm Optimization (PSO) algorithm, suitably adapted for box-constrained optimization. Finally, we report our numerical experience. Preliminary results are provided for a simple hydroelastic problem. Two different numerical tools are involved: a fluid dynamic solver, to simulate the ow around hydrofoils traveling in proximity of the air-water interface, and a simplified torsion-flexional wing model

Optimization Formulations for the Design of Low Embodied Energy Structures Made from Reused Elements

The building sector is one of the major contributors to material resource consumption, greenhouse gas emission and waste production. Load-bearing systems have a particularly large environmental impact because of their material and energy intensive manufacturing process. This paper aims to address the reduction of building structures environmental impacts through reusing structural elements for multiple service lives. Reuse avoids sourcing raw materials and requires little energy for reprocessing. However, to design a new structure reusing elements available from a stock is a challenging problem of combinatorial nature. This is because the structural system layout is a result of the available elements’ mechanical and geometric properties. In this paper, structural optimization formulations are proposed to design truss systems from available stock elements. Minimization of weight, cut-off waste and embodied energy are the objective functions subject to ultimate and serviceability constraints. Case studies focusing on embodied energy minimization are presented for: (1) three roof systems with predefined geometry and topology; (2) a bridge structure whose topology is optimized using the ground structure approach; (3) a geometry optimization to better match the optimal topology from 2 and available stock element lengths. In order to benchmark the energy savings through reuse, the optimal layouts obtained with the proposed methods are compared to weight-optimized solutions made of new material. For these case studies, the methods proposed in this work enable reusing stock elements to design structures embodying up to 71% less energy and hence having a significantly lower environmental impact with respect to structures made of new material

A micro-accelerometer MDO benchmark problem

Many optimization and coordination methods for multidisciplinary design optimization (MDO) have been proposed in the last three decades. Suitable MDO benchmark problems for testing and comparing these methods are few however. This article presents a new MDO benchmark problem based on the design optimization of an ADXL150 type lateral capacitive micro-accelerometer. The behavioral models describe structural and dynamic effects, as well as electrostatic and amplification circuit contributions. Models for important performance indicators such as sensitivity, range, noise, and footprint area are presented. Geometric and functional constraints are included in these models to enforce proper functioning of the device. The developed models are analytical, and therefore highly suitable for benchmark and educational purposes. Four different problem decompositions are suggested for four design cases, each of which can be used for testing MDO coordination algorithms. As a reference, results for an all-in-one implementation, and a number of augmented Lagrangian coordination algorithms are given. © 2009 The Author(s)

Structural shape optimization using Cartesian grids and automatic h-adaptive mesh projection

[EN] We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of thematrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h- adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process.The authors wish to thank the Spanish Ministerio de Economia y Competitividad for the financial support received through the project DPI2013-46317-R and the FPI program (BES-2011-044080), and the Generalitat Valenciana through the project PROMETEO/2016/007.Marco, O.; Ródenas, J.; Albelda Vitoria, J.; Nadal, E.; Tur Valiente, M. (2017). Structural shape optimization using Cartesian grids and automatic h-adaptive mesh projection. Structural and Multidisciplinary Optimization. 1-21. https://doi.org/10.1007/s00158-017-1875-1S121MATLAB version 8.3.0.532 (R2014a) (2014) Documentation. The Mathworks, Inc., Natick, MassachusettsAbel JF, Shephard MS (1979) An algorithm for multipoint constraints in finite element analysis. Int J Numer Methods Eng 14(3):464–467Amestoy P, Davis T, Duff I (1996) An approximate minimum degree ordering algorithm. SIAM J Matrix Anal Appl 17(4):886–905Barth W, Stürzlinger W (1993) Efficient ray tracing for Bezier and B-spline surfaces. Comput Graph 17 (4):423–430Bennett J A, Botkin M E (1985) Structural shape optimization with geometric problem description and adaptive mesh refinement. AIAA J 23(3):459–464Braibant V, Fleury C (1984) Shape optimal design using b-splines. Comput Methods Appl Mech Eng 44 (3):247–267Bugeda G, Oliver J (1993) A general methodology for structural shape optimization problems using automatic adaptive remeshing. Int J Numer Methods Eng 36(18):3161–3185Bugeda G, Ródenas J J, Oñate E (2008) An integration of a low cost adaptive remeshing strategy in the solution of structural shape optimization problems using evolutionary methods. Comput Struct 86(13–14):1563–1578Chang K, Choi K K (1992) A geometry-based parameterization method for shape design of elastic solids. Mech Struct Mach 20(2):215–252Cho S, Ha S H (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip Optim 38(1):53–70Belegundu D, Zhang YMS, Salagame R (1991) The natural approach for shape optimization with mesh distortion control. Tech. rep., Penn State UniversityDavis T A, Gilbert J R, Larimore S, Ng E (2004) An approximate column minimum degree ordering algorithm. ACM Trans Math Softw 30(3):353–376Doctor L J, Torborg J G (1981) Display techniques for octree-encoded objects. IEEE Comput Graph Appl 1(3):29–38Dunning P D, Kim H A, Mullineux G (2011) Investigation and improvement of sensitivity computation using the area-fraction weighted fixed grid FEM and structural optimization. Finite Elem Anal Des 47(8):933–941Düster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197(45-48):3768–3782Escobar J M, Montenegro R, Rodríguez E, Cascón J M (2014) The meccano method for isogeometric solid modeling and applications. Eng Comput 30(3):331–343Farhat C, Lacour C, Rixen D (1998) Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: a numerically scalable algorithm. Int J Numer Methods Eng 43(6):997–1016Fries T P, Omerović S (2016) Higher-order accurate integration of implicit geometries. Int J Numer Methods Eng 106(5):323–371Fuenmayor F J, Oliver J L (1996) Criteria to achieve nearly optimal meshes in the h-adaptive finite element mehod. Int J Numer Methods Eng 39(23):4039–4061Fuenmayor F J, Oliver J L, Ródenas J J (1997) Extension of the Zienkiewicz-Zhu error estimator to shape sensitivity analysis. Int J Numer Methods Eng 40(8):1413–1433García-Ruíz M J, Steven G P (1999) Fixed grid finite elements in elasticity problems. Eng Comput 16 (2):145–164Gill P, Murray W, Saunders M, Wright M (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints. ACM Trans Math Software 10:282–298González-Estrada O A, Nadal E, Ródenas J J, Kerfriden P, Bordas S P A, Fuenmayor F J (2014) Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Comput Mech 53(5):957–976Ha S H, Choi K K, Cho S (2010) Numerical method for shape optimization using T-spline based isogeometric method. Struct Multidiscip Optim 42(3):417–428Haftka R T, Grandhi R V (1986) Structural shape optimization: A survey. Comput Methods Appl Mech Eng 57(1):91–106Haslinger J, Jedelsky D (1996) Genetic algorithms and fictitious domain based approaches in shape optimization. Struc Optim 12:257–264Hughes T J R, Cottrell J A, Bazilevs Y (2005) Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry, and Mesh Refinement. Comput Methods Appl Mech Eng 194:4135–4195Jackins C L, Tanimoto S L (1980) Oct-tree and their use in representing three-dimensional objects. Comput Graphics Image Process 14(3):249–270Kajiya J T (1982) Ray Tracing Parametric Patches. SIGGRAPH Comput Graph 16(3):245–254van Keulen F, Haftka R T, Kim N (2005) Review of options for structural design sensitivity analysis. Part I: linear systems. Comput Methods Appl Mech Eng 194(30-33):3213–3243Kibsgaard S (1992) Sensitivity analysis-the basis for optimization. Int J Numer Methods Eng 34(3):901–932Kikuchi N, Chung K Y, Torigaki T, Taylor J E (1986) Adaptive finite element methods for shape optimization of linearly elastic structures. Comput Methods Appl Mech Eng 57(1):67–89Kim N H, Chang Y (2005) Eulerian shape design sensitivity analysis and optimization with a fixed grid. Comput Methods Appl Mech Eng 194(30–33):3291–3314Kudela L, Zander N, Kollmannsberger S, Rank E (2016) Smart octrees: Accurately integrating discontinuous functions in 3d. Comput Methods Appl Mech Eng 306(1):406–426Kunisch K, Peichl G (1996) Numerical gradients for shape optimization based on embedding domain techniques. Comput Optim 18:95–114Li K, Qian X (2011) Isogeometric analysis and shape optimization via boundary integral. Computer-Aided Design 43(11):1427–1437Lian H, Kerfriden P, Bordas S P A (2016) Implementation of regularized isogeometric boundary element methods for gradient-based shape optimization in two-dimensional linear elasticity. Int J Numer Methods Eng 106 (12):972–1017Liu L, Zhang Y, Hughes T J R, Scott M A, Sederberg T W (2014) Volumetric T-spline Construction using Boolean Operations. Eng Comput 30(4):425–439Marco O, Sevilla R, Zhang Y, Ródenas J J, Tur M (2015) Exact 3D boundary representation in finite element analysis based on Cartesian grids independent of the geometry. Int J Numer Methods Eng 103:445–468Marco O, Ródenas J J, Fuenmayor FJ, Tur M (2017a) An extension of shape sensitivity analysis to an immersed boundary method based on cartesian grids. Computational Mechanics SubmittedMarco O, Ródenas J J, Navarro-Jiménez JM, Tur M (2017b) Robust h-adaptive meshing strategy for arbitrary cad geometries in a cartesian grid framework. Computers & Structures SubmittedMeagher D (1980) Octree Encoding: A New Technique for the Representation, Manipulation and Display of Arbitrary 3-D Objects by Computer. Tech. Rep. IPL-TR-80-11 I, Rensselaer Polytechnic InstituteMoita J S, Infante J, Mota C M, Mota C A (2000) Sensitivity analysis and optimal design of geometrically non-linear laminated plates and shells. Comput Struct 76(1–3):407–420Nadal E (2014) Cartesian Grid FEM (cgFEM): High Performance h-adaptive FE Analysis with Efficient Error Control. Application to Structural Shape Optimization. PhD Thesis. Universitat Politècnica de ValènciaNadal E, Ródenas J J, Albelda J, Tur M, Tarancón J E, Fuenmayor F J (2013) Efficient finite element methodology based on cartesian grids: application to structural shape optimization. Abstr Appl Anal 2013:1–19Najafi A R, Safdari M, Tortorelli D A, Geubelle P H (2015) A gradient-based shape optimization scheme using an interface-enriched generalized FEM. Comput Methods Appl Mech Eng 296:1–17Nguyen V P, Anitescu C, Bordas S P A, Rabczuk T (2015) Isogeometric analysis: An overview and computer implementation aspects. Math Comput Simul 117:89–116Nishita T, Sederberg TW, Kakimoto M (1990) Ray Tracing Trimmed Rational Surface Patches. SIGGRAPH Comput Graph 24(4):337–345Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer-Verlag, New YorkPandey P C, Bakshi P (1999) Analytical response sensitivity computation using hybrid finite elements. Comput Struct 71(5):525–534Parvizian J, Düster A, Rank E (2007) Finite Cell Method: h- and p- Extension for Embedded Domain Methods in Solid Mechanics. Comput Mech 41(1):121–133Peskin C S (1977) Numerical Analysis of Blood Flow in the Heart. J Comput Phys 25:220–252Poldneff M J, Rai I S, Arora J S (1993) Implementation of design sensitivity analysis for nonlinear structures. AIAA J 31(11):2137–2142Powell M (1983) Variable metric methods for constrained optimization. In: Bachem A, Grotschel M, Korte B (eds) Mathematical Programming: The State of the Art, Springer, Berlin, Heidelberg, pp 288–311Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Eng 199(29–32):2059–2071Riehl S, Steinmann P (2014) An integrated approach to shape optimization and mesh adaptivity based on material residual forces. Comput Methods Appl Mech Eng 278:640–663Riehl S, Steinmann P (2016) On structural shape optimization using an embedding domain discretization technique. Int J Numer Methods Eng 109(9):1315–1343Ródenas J J, Tarancón J E, Albelda J, Roda A, Fuenmayor F J (2005) Hierarchical Properties in Elements Obtained by Subdivision: a Hierarquical h-adaptivity Program. In: Díez P, Wiberg N E (eds) Adaptive Modeling and Simulation, p 2005Ródenas J J, Corral C, Albelda J, Mas J, Adam C (2007a) Nested domain decomposition direct and iterative solvers based on a hierarchical h-adaptive finite element code. In: Runesson K, Díez P (eds) Adaptive Modeling and Simulation 2007, Internacional Center for Numerical Methods in Engineering (CIMNE), pp 206–209Ródenas J J, Tur M, Fuenmayor F J, Vercher A (2007b) Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique. Int J Numer Methods Eng 70(6):705–727Ródenas J J, Bugeda G, Albelda J, Oñate E (2011) On the need for the use of error-controlled finite element analyses in structural shape optimization processes. Int J Numer Methods Eng 87(11):1105–1126Schillinger D, Ruess M (2015) The finite cell method: A review in the context of higher-order structural analysis of cad and image-based geometric models. Arch Comput Meth Eng 22(3):391– 455Sevilla R, Fernández-Méndez S, Huerta A (2011a) 3D-NURBS-enhanced Finite Element Method (NEFEM). Int J Numer Methods Eng 88(2):103–125Sevilla R, Fernández-Méndez S, Huerta A (2011b) Comparison of High-order Curved Finite Elements. Int J Numer Methods Eng 87(8):719–734Sevilla R, Fernández-Méndez S, Huerta A (2011c) NURBS-enhanced Finite Element Method (NEFEM): A Seamless Bridge Between CAD and FEM. Arch Comput Meth Eng 18(4):441–484Sweeney M, Bartels R (1986) Ray tracing free-form b-spline surfaces. IEEE Comput Graph Appl 6(2):41–49Toth D L (1985) On Ray Tracing Parametric Surfaces. SIGGRAPH Comput Graph 19(3):171–179Tur M, Albelda J, Nadal E, Ródenas J J (2014) Imposing dirichlet boundary conditions in hierarchical cartesian meshes by means of stabilized lagrange multipliers. Int J Numer Methods Eng 98(6):399–417Tur M, Albelda J, Marco O, Ródenas J J (2015) Stabilized Method to Impose Dirichlet Boundary Conditions using a Smooth Stress Field. Comput Methods Appl Mech Eng 296:352–375Yao T, Choi KK (1989) 3-d shape optimal design and automatic finite element regridding. Int J Numer Methods Eng 28(2):369–384Zhang L, Gerstenberger A, Wang X, Liu W K (2004) Immersed Finite Element Method. Comput Methods Appl Mech Eng 293(21):2051–2067Zhang Y, Wang W, Hughes T J R (2013) Conformal Solid T-spline Construction from Boundary T-spline Representations. Comput Mech 6(51):1051–1059Zienkiewicz O C, Zhu J Z (1987) A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis. Int J Numer Methods Eng 24(2):337–35