Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations

Gómez, Héctor; Hughes, Thomas J.R.; Nogueira, Xesús; Calo, Victor M.

doi:10.1016/j.cma.2010.02.010

Cited by 218 publications

(149 citation statements)

References 51 publications

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“…In spite of the increased cost of highercontinuous spaces, they have proven very popular and useful. For example, higher-continuous spaces have allowed the solution of higher-order partial di↵erential equations with elegance [7,28,29,51,16,15] as well as several non-linear problems of engineering interest [31,6,30,5,20,11,9,4]. Thus, e cient multi-frontal solvers for higher-continuous spaces are important.…”

Section: Introductionmentioning

confidence: 99%

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Woźniak

Kuźnik

Paszyński

et al. 2014

Computers & Mathematics with Applications

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In this paper we present computational cost estimates for parallel shared memory isogeometric multi-frontal solver. The estimates show that the ideal isogeometric shared memory parallel direct solver scales as O(p 2 log(N/p)) for one dimensional problems, O(Np 2 ) for two dimensional problems, and O(N 4/3 p 2 ) for three dimensional problems, where N is the number of degrees of freedom, and p is the polynomial order of approximation. The computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver, being the latest equal to O(Np 2 ) for the one dimensional case, O(N 1.5 p 3 ) for the two dimensional case, and O(N 2 p 3 ) for the three dimensional case. The shared memory version significantly reduces both the scalability in terms of N and p. Theoretical estimates are compared with numerical experiments performed with linear, quadratic, cubic, quartic, and quintic B-splines, in one and two spatial dimensions.Keywords: isogeometric finite element method, multi-frontal direct solver, computational cost, NVIDIA CUDA GPU Preprint submitted to Computers & Mathematics with ApplicationsMarch 27, 20141. Introduction Classical higher order finite element methods (FEM) [17,18] maintain only C 0 -continuity at the element interfaces, while isogeometric analysis (IGA) utilizes B-splines as basis functions, and thus, it delivers C k global continuity [14]. The higher continuity obtained across elements allows IGA to attain optimal convergence rates for any polynomial order, while using fewer degrees of freedom [3,1]. Nevertheless, this reduced count in the number of degrees of freedom may not immediately correlate with a computational cost reduction, since solution time per degree of freedom augments as the continuity is increased [10,13]. In spite of the increased cost of highercontinuous spaces, they have proven very popular and useful. For example, higher-continuous spaces have allowed the solution of higher-order partial di↵erential equations with elegance [7,28,29,51,16,15] as well as several non-linear problems of engineering interest [31,6,30,5,20,11,9,4]. Thus, e cient multi-frontal solvers for higher-continuous spaces are important.The multi-frontal solver is one of the state-of-the art algorithm for solving linear systems of equations [22,26]. It is a generalization of the frontal solver algorithm proposed in [33,21]. The multi-frontal algorithm constructs an assembly tree based on the analysis of the connectivity data or the geometry of the computational mesh. Finite elements are joint into pairs and fully assembled unknowns are eliminated within frontal matrices associated to multiple branches of the tree. The process is repeated until the root of the assembly tree is reached. Finally, the common interface problem is solved and partial backward substitutions are recursively called on the assembly tree.There exist parallel versions of the multi-frontal direct solver algorithm targeting distributed-memory, share...

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Section: Introductionmentioning

confidence: 99%

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Woźniak

Kuźnik

Paszyński

et al. 2014

Computers & Mathematics with Applications

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“…Therefore, this model opens the possibility to solve vexing problems in computational mechanics such as for example, cavitation, film and nucleate boiling or the phasechange-driven implosion of thin structures (Bueno et al, 2014). The Navier-Stokes-Korteweg equations are rather new in the field, but there are interesting works of theoretical (Dunn and Serrin, 1986) and computational nature (Gomez et al, 2010b;Liu et al, 2013Liu et al, , 2015Tian et al, 2015;Giesselmann et al, 2014;Jamet et al, 2001). Our point of departure to derive the Navier-Stokes-Korteweg equations are classical balance laws for mass, linear momentum, angular momentum and energy, which may be written as followṡ…”

Section: Mechano-biological Mixtures: Phase-field Tumor Growth Theorymentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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“…The first instantiations of isogeometric analysis were based on Non-Uniform Rational B-Splines (NURBS), and to date, NURBS-based isogeometric analysis has achieved great success in a number of application areas including complex flow problems [5,7] and phasefield models [30,32]. Recently, isogeometric analysis technologies based on other classes of basis functions have been developed.…”

Section: Isogeometric Analysismentioning

confidence: 99%

“…It has been shown that isogeometric analysis possesses several advantages over traditional finite element methods in terms of approximation accuracy [19,24], robustness [49], and mesh generation [4,37]. Additionally, it has been successfully applied to various fluid problems [5,6,25] and phase-field models [30,32]. For a general discussion of isogeometric analysis, the reader is referred to [18].…”

Section: Semi-discrete Formulationmentioning

confidence: 99%

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

Liu¹,

Gómez²,

Evans³

et al. 2012

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We propose a new methodology for the numerical solution of the isothermal Navier-Stokes-Korteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws. AbstractWe propose a new methodology for the numerical solution of the isothermal Navier-StokesKorteweg equations. Our methodology is based on a semi-discrete Galerkin method invoking functional entropy variables, a generalization of classical entropy variables, and a new time integration scheme. We show that the resulting fully discrete scheme is unconditionally stable-in-energy, second-order time-accurate, and mass-conservative. We utilize isogeometric analysis for spatial discretization and verify the aforementioned properties by adopting the method of manufactured solutions and comparing coarse mesh solutions with overkill solutions. Various problems are simulated to show the capability of the method. Our methodology provides a means of constructing unconditionally stable numerical schemes for nonlinear non-convex hyperbolic systems of conservation laws.

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Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations

Cited by 218 publications

References 51 publications

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Computational Phase‐Field Modeling

Functional Entropy Variables: A New Methodology for Deriving Thermodynamically Consistent Algorithms for Complex Fluids, with Particular Reference to the Isothermal Navier-Stokes-Korteweg Equations

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