Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Bartoň, Michael; Puzyrev, Vladimir; Deng, Quanling; Calo, Victor M.

doi:10.1016/j.cam.2019.112626

Cited by 14 publications

(10 citation statements)

References 30 publications

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“…The integration of B-spline basis functions is needed in IGA, for example, in the computation of the mass and stiffness matrix elements Bartoň et al (2017). In order to use the Gaussian quadrature method Atkinson (2008) for precise computation of the integral over the B-spline basis functions, the B-spline basis functions need to be split into polynomial or rational (in the NURBS case) functions.…”

Section: Subdivision Into Bézier Trimmed Trivariatesmentioning

confidence: 99%

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Massarwi

Antolín

Elber

2019

Computer Aided Geometric Design

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3D objects, modeled using Computer Aided Geometric Design (CAGD) tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain.A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task Xu et al. (2017). In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process, in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed Bézier trivariates, at all its internal knots. Then, each trimmed Bézier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples of the algorithm on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.are the control points of T , and B i,d is the i'th univariate B-spline basis functions of degree d.3D geometric objects, generated with contemporary computer aided geometric design (CAGD) systems, are almost solely exploiting a boundary representation (B-rep), where these objects' boundaries are represented as trimmed parametric surfaces. In recent years, with recent developments of additive manufacturing and 3D printing as well as analysis, the demand for a full volumetric representation (V-rep) is increasing. Volumetric modeling of the inside of the object, as oppose to only its boundary (B-rep), can be used to describe different volumetric properties such as materials or stresses, in scalar, vector or tensor fields. Developments in physical analysis, isogeometric analysis (IGA) in specific Cottrell et al. (2009), employs a parametrization of the object's volume. In IGA, the analysis is performed by integrating in the same spline spaces that describe the geometry.Tensor product trivariates are limited to a cuboid topology, making them difficult to use in creating general 3D objects, having an arbitrar...

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Section: Subdivision Into Bézier Trimmed Trivariatesmentioning

confidence: 99%

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Massarwi

Antolín

Elber

2019

Computer Aided Geometric Design

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“…Computing F involves the evaluation of G and/or its derivatives, cf. (4). Each component of G, or of a derivative of G, is a spline function and can be evaluated by the algorithm in Section 6.2.…”

Section: Evaluation Of U ∈ Span φmentioning

confidence: 99%

“…Authors from the same group have then further reduced the computational complexity by weighted quadrature, cf. the publication by Calabrò, Sangalli, and Tani [10] and the related publication [4]. Here, the idea is to reduce the number of quadrature points by setting up appropriately adjusted quadrature rules.…”

Section: Introductionmentioning

confidence: 99%

Sum factorization techniques in Isogeometric Analysis

Bressan

Takacs

2019

Computer Methods in Applied Mechanics and Engineering

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The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order p. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as p d+2 instead of p 2d+1 as previously achieved.

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“…Recently, machine learning (ML) has been proven effective and successful in many fields such as medical diagnoses, image and speech recognition, financial services, autopilot in automotive scenarios, and many other engineering and medical applications 16‐18 . Computational engineering and mechanics are no exception 19‐21 . Several data‐driven approaches have been developed to capture the thermal conductivity of composites, 22 the elastic properties of composites, 23 the anisotropic hyperelasticity, 24 the plastic response of different systems, 25‐28 the thermo‐viscoplastic modeling of solidification, 28,29 the failure of composites, 30 the fatigue of materials, 31 the effect of flexoelectricity on nanostructures, 32 the properties of phononic crystals, 33 and so forth.…”

Section: Introductionmentioning

confidence: 99%

Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

Abueidda

Korić

2021

Numerical Meth Engineering

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Deep learning (DL) and the collocation method are merged and used to solve partial differential equations (PDEs) describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo‐Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data‐driven models. The DL model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the DCM is potentially a promising standalone technique to solve PDEs involved in the deformation of materials and structural systems as well as other physical phenomena.

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Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines

Cited by 14 publications

References 30 publications

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Sum factorization techniques in Isogeometric Analysis

Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

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