94 research outputs found

    Metodología jerárquica h adaptativa basada en subdivisión de elementos

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    [EN] This paper presents a hierarchical h adaptive methodology for Finite Element Analysis based on the hierarchical relations between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition the terms involved in the evaluation of element stiffness matrices of parent and child elements are related by a constant which is a function of the element sizes ratio (scaling factor). These relations have been the basis for the development of a hierarchical h adaptivity methodology based on element subdivision and the use of multipoint-constraints to ensure C0 continuity. The use of a hierarchical data structure significantly reduces the amount of calculations required for the mesh refinement, the evaluation of the global stiffness matrix, element stresses and element error estimation. The data structure also produces a natural reordering of the global stiffness matrix that improves the behaviour of the Cholesky factorization.[ES] En este artículo se presenta una metodología h adaptativa para el Análisis por Elementos Finitos basada en las relaciones jerárquicas entre elementos padre e hijo que surgen si estos elementos son geométricamente similares. Bajo esta condición de similitud, los términos resultantes de la evaluación de las matrices de rigidez de elementos padre e hijo están relacionados por una constante que es una función de la relación de tamaños de elemento (factor de escala). Estas relaciones han sido la base para el desarrollo de una metodología jerárquica h adaptativa basada en la subdivisión de elementos y el uso de restricciones multipunto para asegurar la continuidad C0 . El uso de una estructura de datos jerárquica reduce significativamente la cantidad de cálculos requeridos para el refinamiento de la malla, la evaluación de la matriz de rigidez global, las tensiones de los elementos y la estimación del error del elemento. La estructura de datos también produce un reordenamiento natural de la matriz de rigidez global que mejora el comportamiento de la factorización de Cholesky.The authors wish to thank the Spanish Ministerio de Economía y Competitividad for the fiancial support received through the project DPI2013-46317-R and the Generalitat Valenciana through the project PROMETEO/2016/007. The support of the Universidad Politécnica de Valencia is also acknowledged. The authors also want to thank Ana Ródenas’s help in the translation of this paper.Ródenas, J.; Albelda Vitoria, J.; Tur Valiente, M.; Fuenmayor Fernández, F. (2017). A hierarchical h adaptivity methodology based on element subdivision. Revista UIS Ingenierías. 16(2):263-280. https://doi.org/10.18273/revuin.v16n2-2017024S26328016

    Metodología jerárquica h-adaptativa basada en subdivisión de elementos

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    This paper presents a hierarchical h adaptive methodology for Finite Element Analysis based on the hierarchical relations between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition the terms involved in the evaluation of element stiffness matrices of parent and child elements are related by a constant which is a function of the element sizes ratio (scaling factor). These relations have been the basis for the development of a hierarchical h adaptivity methodology based on element subdivision and the use of multi-point-constraints to ensure C0 continuity. The use of a hierarchical data structure significantly reduces the amount of calculations required for the mesh refinement, the evaluation of the global stiffness matrix, element stresses and element error estimation. The data structure also produces a natural reordering of the global stiffness matrix that improves the behaviour of the Cholesky factorization.Este artículo presenta una metodología h-adaptativa jerárquica para el análisis de elementos finitos basado en las relaciones jerárquicas entre los elementos padre e hijo que aparecen si estos elementos son geométricamente similares. Bajo esta condición de similitud, los términos implicados en la evaluación de las matrices de rigidez de elementos de los elementos padre e hijo están relacionados por una constante que es una función de la relación de tamaños de elemento (factor de escala). Estas relaciones han sido la base para el desarrollo de una metodología h-adaptativa jerárquica basada en la subdivisión de elementos y el uso de restricciones multipunto para asegurar la continuidad C0. El uso de una estructura jerárquica de datos reduce significativamente la cantidad de cálculos requeridos para el refinamiento de la malla, la evaluación de la matriz de rigidez global, las tensiones de los elementos y la estimación del error del elemento. La estructura de datos también produce un reordenamiento natural de la matriz de rigidez global que mejora el comportamiento de la factorización de Cholesky

    Sparse approximations in spatio-temporal point-process models

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    Analysis of spatio-temporal point patterns plays an important role in several disci-plines, yet inference in these systems remains computationally challenging due to the high resolution modelling generally required by large data sets and the analytically in-tractable likelihood function. Here, we exploit the sparsity structure of a fully-discretised log-Gaussian Cox process model by using expectation constrained approximate inference. The resulting family of expectation propagation algorithms scale well with the state di-mension and the length of the temporal horizon with moderate loss in distributional accu-racy. They hence provide a flexible and faster alternative to both the filtering-smoothing type algorithms and the approaches which implement the Laplace method or expectation propagation on (block) sparse latent Gaussian models. We demonstrate the use of the proposed method in the reconstruction of conflict intensity levels in Afghanistan from a WikiLeaks data set

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

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    [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

    An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

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    We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices

    Performance Modeling and Prediction for the Scalable Solution of Partial Differential Equations on Unstructured Grids

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    This dissertation studies the sources of poor performance in scientific computing codes based on partial differential equations (PDEs), which typically perform at a computational rate well below other scientific simulations (e.g., those with dense linear algebra or N-body kernels) on modern architectures with deep memory hierarchies. We identify that the primary factors responsible for this relatively poor performance are: insufficient available memory bandwidth, low ratio of work to data size (good algorithmic efficiency), and nonscaling cost of synchronization and gather/scatter operations (for a fixed problem size scaling). This dissertation also illustrates how to reuse the legacy scientific and engineering software within a library framework. Specifically, a three-dimensional unstructured grid incompressible Euler code from NASA has been parallelized with the Portable Extensible Toolkit for Scientific Computing (PETSc) library for distributed memory architectures. Using this newly instrumented code (called PETSc-FUN3D) as an example of a typical PDE solver, we demonstrate some strategies that are effective in tolerating the latencies arising from the hierarchical memory system and the network. Even on a single processor from each of the major contemporary architectural families, the PETSc-FUN3D code runs from 2.5 to 7.5 times faster than the legacy code on a medium-sized data set (with approximately 105 degrees of freedom). The major source of performance improvement is the increased locality in data reference patterns achieved through blocking, interlacing, and edge reordering. To explain these performance gains, we provide simple performance models based on memory bandwidth and instruction issue rates. Experimental evidence, in terms of translation lookaside buffer (TLB) and data cache miss rates, achieved memory bandwidth, and graduated floating point instructions per memory reference, is provided through accurate measurements with hardware counters. The performance models and experimental results motivate algorithmic and software practices that lead to improvements in both parallel scalability and per-node performance. We identify the bottlenecks to scalability (algorithmic as well as implementation) for a fixed-size problem when the number of processors grows to several thousands (the expected level of concurrency on terascale architectures). We also evaluate the hybrid programming model (mixed distributed/shared) from a performance standpoint
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