isogeometric analysis for 2d and 3d mhd subproblems...

IGA 2018: Integrating Design and Analysis

October 10-12, 2018, Austin, TX

ISOGEOMETRIC ANALYSIS FOR 2D AND 3D MHD

SUBPROBLEMS: SPECTRAL SYMBOL AND FAST ITERATIVE

SOLVERS

Carla Manni1, Mariarosa Mazza2, Ahmed Ratnani2, Stefano Serra-Capizzano3, and Hendrik

Speleers1

1University of Rome “Tor Vergata”, Rome, Italy {manni,speleers}@axp.mat.uniroma2.it

2Max Planck Institute for Plasma Physics, Garching, Germany {mariarosa.mazza,ahmed.ratnani}@ipp.mpg.de

3University of Insubria, Como, Italy [email protected]

ABSTRACT

In plasma physics, magnetohydrodynamics (MHD) is used to study the macroscopic behavior of

plasma. In this talk, we focus on a parameter-dependent curl-div subproblem and we discretize it

using isogeometric analysis based on tensor product B-splines. The involved coefficient matrices

can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here.

In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is

proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the

critical dependence on the different physical and approximation parameters. Second, we exploit

such spectral information to design fast iterative solvers for the corresponding linear systems. For

the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz

sequences [1], while for the second we use a combination of multigrid techniques and

preconditioned Krylov solvers. Several numerical tests are provided both for the study of the

spectral problem and for the solution of the corresponding linear systems.

REFERENCES

[1] C. Garoni, and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and

Applications - Vol I, Springer Monographs, 2017.

IGA 2018: Integrating Design and Analysis October 10-12, 2018, Austin, TX

MATRIX-FREE WEIGHTED QUADRATURE FOR A COMPUTATIONALLY EFFICIENT ISOGEOMETRIC K-METHOD

G. Sangalli1, M. Tani2

1 Università di Pavia [email protected]

2 Università di Pavia

[email protected]

ABSTRACT One of the distinguishing features of Isogeometric Analysis (IGA) is the possibility of using high- degree high-regularity splines (the so-called k-refinement) as they deliver higher accuracy per degree-of-freedom in comparison to C0 finite elements. Unfortunately, if the implementation is done following the approaches that are standard in the context of C0 finite elements, the computational cost increases dramatically with the spline degree p. This is true both for the formation of the linear system and for its numerical solution. As a consequence, the use of k-refinement is often unfeasible for practical problems, where quadratic or cubic splines are typically preferred. In this talk we discuss a matrix-free implementation, recently proposed in [1] for scalar elliptic problems, which is very benecial in terms of both memory and computational cost. In particular, the memory required is practically independent of p and the cost depends on p only mildly. Two key ingredients that contribute to achieve this result are the preconditioner discuessed in [2], which is robust with respect to both the mesh size h and the spline degree p, and weighted quadrature, a novel quadrature approach first presented in [3], where the number of quadrature points required is roughly independent of p. The numerical experiments show that, with the new implementation, the k-refinement becomes appealing from the computational point of view. Indeed, increasing the degree and continuity leads to orders of magnitude higher computational efficiency with respect to standard approaches. REFERENCES [1] G. Sangalli and M. Tani, Matrix-free weighted quadrature for a computationally efficient isogeometric k-method, Comput. Methods Appl. Mech. Engrg., 338, 117-133, 2018. [2] G. Sangalli and M. Tani, Isogeometric preconditioners based on fast solvers for the Sylvester

equation, SIAM J. Sci. Comput., 38, A3644-A3671, 2016. [3] F. Calabrò, G. Sangalli and M. Tani, Fast formation of isogeometric Galerkin matrices by

weighted quadrature, Comput. Methods Appl. Mech. Engrg., 316, 606-622, 2017.



QUADRATURE RULES FOR GALERKIN IGA BEM BASED ON

SPLINE QUASI-INTERPOLATION

Antonella Falini1

1 INdAM, c/o Department of Mathematics and Computer Science, University of Florence, [email protected]

ABSTRACT

In the context of the isogeometric Boundary Element Method (BEM) the presence of singular

kernels motivate the necessity to accurately evaluate the involved integrals. To this end, we

designed suitable quadrature rules based on spline quasi-interpolation (QI) [1]. We developed two

algorithmic procedures specific for the isogeometric settings. In particular the QI already

introduced in [2] for regular integrals is employed. The new quadrature formulas are derived

combining the use of the modified moments and splines product rules [3]. In this talk several

numerical examples will be provided comparing the achieved accuracy with respect to standard

and newest techniques, validating the derived theoretical results for the order of convergence.

Finally, some applications to elliptic problems using a variational Galerkin method will be

presented in the set up framework of the isogeometric BEM [4-5-6].

This work is based on joint research with F. Calabrò, C. Giannelli, T. Kanduč, M. L. Sampoli, A.

Sestini.

REFERENCES

[1] F. Calabrò, A. Falini, M.L. Sampoli, and A. Sestini, Efficient quadrature rules based on spline

quasi-interpolation for application to IGA-BEMs, Journal of Computational and Applied

Mathematics, 338, 153-167, 2018.

[2] F. Mazzia, and A. Sestini, Quadrature formulas descending from BS Hermite spline quasi-

interpolation, Journal of Computational and Applied Mathematics, 236, 4015-4118, 2012.

[3] K. Mørken. Some identities for products and degree raising of splines, Constructive

Approximation, 7, 195-208, 1991.

[4] A. Aimi, F. Calabrò, M. Diligenti, M.L. Sampoli, G. Sangalli, and A. Sestini, Efficient

assembly based on B-spline tailored quadrature rules for the IgA-SGBEM, Computer Methods in

Applied Mechanics and Engineering, 331, 327-342, 2018.

[5] A. Falini and T. Kanduč, A study on spline quasi-interpolation based quadrature rules for the

isogeometric Galerkin BEM, preprint in Proceedings Springer INdAM Series.

[6] A. Falini, C. Giannelli, T. Kanduč, M.L. Sampoli, and A. Sestini, An adaptive IgA-BEM with

hierarchical B-splines based on quasi-interpolation quadrature schemes, preprint in International

Journal for Numerical Methods in Engineering.


P-MULTIGRID BASED SOLVERS FOR IGA

Roel Tielen1, Matthias Möller1, and Kees Vuik1

1 Delft University of Technology{r.p.w.m.tielen, m.moller, c.vuik}@tudelft.nl

ABSTRACTOver the years, Isogeometric Analysis (IgA) [1] has shown to be a successful alternative to theFinite Element Method, both in academia and industrial applications. However, solving theresulting linear systems efficiently remains a challenging task. For instance, the condition numberof the stiffness matrix scales quadratically with the mesh width h, but, in contrast to standard FiniteElements, exponentially with the order of the approximation p [2]. The performance of (standard)iterative solvers thus decreases fast for larger values of p.

In this talk we present a solution strategy for IgA discretizations that is based on p-multigridtechniques used both as a solver and as a preconditioner in an outer Krylov subspace iterationmethod. In contrast to (geometric) h-multigrid methods, where a hierarchy of coarser and finermeshes is constructed, the approach makes use of a hierarchy of B-spline based discretizations ofdifferent approximation orders. The ‘coarse grid’ correction is determined at level p = 1, whichenables us to use well established solution techniques developed for low-order Lagrange finiteelements. Since both prolongation and restriction operators are defined as mappings betweenarbitrary spline spaces, combined coarsening in both h and p is possible, leading to a flexible hp-multigrid.

We present numerical results obtained with both p- and hp-multigrid methods for differentbenchmark problems on non-trivial geometries. It follows from a spectral analysis, that the coarsegrid correction and the smoothing procedure complement each other quite well for both methods.In particular we investigate the performance of non-standard smoothers to further improve theconvergence rates. Our preliminary results indicate that p-multigrid based methods have thepotential to efficiently solve IgA discretizations.

REFERENCES

[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, ”Isogeometric Analysis: CAD, Finite Elements,

NURBS, Exact Geometry and Mesh Refinement”, Computer Methods in Applied Mechanics

and Engineering, 194, 4135 − 4195, 2005

[2] K.P.S. Gahalaut, J.K. Kraus, S.K. Tomar. ”Multigrid methods for isogeometric discretization”,

Computer Methods in Applied Mechanics and Engineering, 253, 413 − 425, 2013



EFFICIENT QUADRATURE FOR HIGH ORDER ISOGEOMETRIC METHOD

F. Calabrò1, G. Sangalli2, and M. Tani2

1 U. di Cassino - Italy [email protected]

2 U. di Pavia - Italy {giancarlo.sangalli mattia.tani}@unipv.it

ABSTRACT

In this talk, we present recent results on the assembly of the linear system arising in the Galerkin

isogeometric method.

The main interest are the cases where the degree of the approximation is raised, so that the

computational cost in assembling become challenging.

Key ingredients are the application of weighted quadrature and row loops. These modifications

demand for a change of paradigm the existing fem-based codes. Also new results on the

application of Generalized Gaussian quadrature will be presented.

REFERENCES

[1] F. Calabrò, G. Sangalli and M. Tani, Fast formation of isogeometric Galerkin matrices by

weighted quadrature, Comput. Methods Appl. Mech. Engrg., 316, 606-622, (2017).

[2] A. Aimi, F. Calabrò, M. Diligenti, M.L. Sampoli, G. Sangalli and A. Sestini, Efficient

assembly based on B-spline tailored quadrature rules for the IgA-SGBEM, Comput. Methods Appl.

Mech. Engrg., 331, 327-342, (2018).

[3] P. Antolin, A. Buffa, F. Calabrò, M. Martinelli and G. Sangalli, Efficient matrix computation

for tensor-product isogeometric analysis: The use of sum factorization, Comput. Methods Appl.

Mech. Engrg., 285, 817-828, (2015).



PARALLEL HYBRID MEMORY ISOGEOMETRIC

ALTERNATING DIRECTIONS COLLOCATION METHOD

M. Woźniak1, M. Łoś2, and M. Paszyński3

1 AGH University of Science of Technology

al. Mickiewicza 30, 30-059 Kraków, Poland {macwozni,los,paszynsk}@agh.edu.pl

ABSTRACT

Simulations of the oil extraction process by hydraulic fracturing require the solution of non-linear

systems of partial differential equations describing the non-linear flow in heterogeneous media [1].

For this purpose, the isogeometric analysis and linear computational cost solver utilizing the

Kronecker product structure of the matrix is used [2, 3]. Unfortunately, the cost of factorization is

dominated by the cost of numerical integration of higher-order B-spline functions, and this cost

dominates the entire calculation process (e.g. in the case of parallel distributed memory computing,

this cost represents over 94 percent of simulation time [4]). We propose to replace the

isogeometric finite element method with the isogeometric collocation method [5], which allows

reducing the computational cost by two orders of magnitude. We present a parallel hybrid memory

implementation of the isogeometric collocation method. We test it on PROMETHEUS Linux

cluster from CYFRONET supercomputing center in Kraków. We use up to 256 nodes with 24

cores each. We show almost perfect scalability of the resulting parallel code.

ACKNOWLEDGEMENT

This work was supported by Dean's Grant from Faculty of Computer Science, Electronics and

Telecommunication, AGH University of Science and Technology, Kraków, Poland.

REFERENCES

[1] M. Alotaibi, V.M. Calo, Y. Efendiev, J. Galvis, and M. Ghommem, Global-Local Nonlinear

Model Reduction for Flows in Heterogeneous Porous Media, Computer Methods in Applied

Mechanics, and Engineering, 292 (1) 122-137, 2015.

[2] B. Barabasz, M. Łoś, M. Woźniak, L. Siwik, and S. Barrett, Coupled isogeometric Finite

Element Method and Hierarchical Genetic Strategy with balanced accuracy for solving

optimization inverse problem, Procedia Computer Science, 108, 828-837, 2017.

[3] M. Łoś, M. Woźniak, M. Paszyński, A.M. Hassan, and K. Pingali, IGA-ADS: Isogeometric

analysis FEM using ADS solver, Computer & Physics Communications 217, 99-116, 2017.

[4] M. Woźniak, M. Łoś, M. Paszynski, and V. M. Calo, Parallel fast isogeometric solvers for

explicit dynamics, Computing and Informatics, 36(2) 423-448, 2017.

[5] A. Reali, and T. J. R. Hughes, An Introduction to Isogeometric Collocation Methods. In: Beer

G., Bordas S. (eds) Isogeometric Methods for Numerical Simulation. CISM International

Centre for Mechanical Sciences, 561, Springer, Vienna, 2015.

IGA 2018: Integrating Design and AnalysisOctober 10-12, 2018, Austin, TX

LOW-RANK APPROXIMATION FOR ISOGEOMETRICANALYSIS

Angelos Mantzaflaris

Institute of Applied Geometry, Johannes Kepler UniversityAltenberger Str. 69, 4040 Linz, Austria

[email protected]

ABSTRACTLow-rank approximation techniques play an integral role in reducing the complexity and restoringcomputational tractability in areas such as optimization, sensitivity analysis, inverse problems, bio-logy and chemistry. In essence, these methods provide ways to recover structure in the parametersor carefully select a few, so that the evaluation of the, so called, reduced model, on large data gridsbecomes tractable. This is done by a suitable projection of the function on a lower-dimensional man-ifold of tensors, whose dimension is called the rank of the tensor. Different notions of ranks and thecorresponding low-rank approximation formats have been introduced, having different approxima-tion and computational complexity properties. In this talk we employ some popular low-rank formatsand elaborate on their use in isogeometric analysis. In particular, the spline discretisations employedin isogeometric analysis possess a global tensor product structure which can be used in several waysto reduce the complexity of the numerical integration and overall solution process [1, 2, 3, 4]. Theexploitation of this tensor product structure enables us to deal with the computational disadvantagesstemming from the increased polynomial degrees and the larger support of the basis functions.

REFERENCES

[1] A. Mantzaflaris, B. Juettler, B. Khoromskij, and U. Langer. Matrix generation in isogeometricanalysis by low rank tensor approximation. In J.-D. Boissonnat, A. Cohen, O. Gibaru, C. Gout,T. Lyche, M.-L. Mazure, and L. L. Schumaker, editors, Curves and Surfaces, volume 9213 ofLNCS, pages 321–340. Springer, 2015.

[2] A. Mantzaflaris, B. Juettler, B. N. Khoromskij, and U. Langer. Low rank tensor methods ingalerkin-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineer-ing, 316:1062 – 1085, 2017. Special Issue on Isogeometric Analysis: Progress and Challenges.

[3] F. Scholz, A. Mantzaflaris, and B. Juettler. Partial tensor decomposition for decoupling isoge-ometric galerkin discretizations. Computer Methods in Applied Mechanics and Engineering,336:485 – 506, 2018.

[4] F. Scholz, A. Mantzaflaris, and I. Toulopoulos. Low-rank space-time decoupled isogeometricanalysis for parabolic problems with varying coefficients. Computational Methods in AppliedMathematics, 2018. (20 pages, to appear).



AN ALGORITHM FOR TENSOR PRODUCT APPROXIMATION

OF THREE-DIMENSIONAL MATERIAL DATA FOR IMPLICIT

DYNAMICS SIMULATIONS

Krzysztof Podsiadło1, Marcin Łoś

1, Leszek Siwik

1, Maciej Woźniak

1 and Maciej Paszyński

1

1AGH University of Science and Technology, Krakow, Poland [podsiadlo,los,siwik,macwozni]@agh.edu.pl

ABSTRACT

In this paper we discuss the possible extension of explicit simulation of non-linear flow in

heterogenous media into semi-implicit case [1]. Namely, we consider the simulation of hydraulic

fracturing related to the oil extraction process. Our goal is to reduce the number of time steps in

the explicit simulations [2]. In the weak form

(1)

we treat the static permeability in implicit way and dynamic permeability

in explicit way. We perform tensor product approximation of static permeability

(2)

following the idea presented in [3]. Next, we perform a single time step semi-implicit computation

in parallel, for each mode of the tensor product approximation

separatelly. We collect the results and get the final

soluiton by computing an average from the parallel instances with different modes.

Figure 1. Map of the formation permeability (left panel) Error of the approximation of material

data with a series of 100 tensor products (middle panel) Snapshot on the pressure computed by the

simulator (right panel)

Acknowledgement. National Science Centre grant no. 2014/15/N/ST6/04662

REFERENCES

[1] M. Alotaibi, V.M. Calo, Y. Efendiev, J. Galvis, and M. Ghommem, Global-Local Nonlinear

Model Reduction for Flows in Heterogeneous Porous Media, Computer Methods in Applied

Mechanics and Engineering 292 (1) 122-137, 2015.

[2] M. Łoś, M. Woźniak, M. Paszyński, A. Lenharth, M. Amber-Hassan and K. Pingali, IGA-

ADS: isogeometric analysis FEM using ADS solver, Computer Physics Communication, 217,

99–116, 2017.

[3] K. Podsiadło, M. Łoś, L. Siwik , and M. Woźniak, An Algorithm for Tensor Product

Approximation of Three-Dimensional Material Data for Implicit Dynamics Simulations,

Lecture Notes in Computer Science, 10861, 156–168, 2018.


AUTOMATIC VARIATIONALLY STABLE ANAYLSIS FOR FE COMPUTATIONS BASED ON THE DPG FRAMEWORK Albert Romkes1, Victor M. Calo2, Maciej Paszynski3, Marcin Los3, and Eirik Valseth1

1 Department of Mechanical Engineering, South Dakota School of Mines & Technology, Rapid City, SD, USA.

[email protected] [email protected]

2 Applied Geology Department, Curtin University, Perth, Australia.

[email protected]

3 Department of Computer Science, AGH University of Technology, Krakow, Poland. [email protected]

[email protected]

ABSTRACT We introduce an automatic variationally stable (AVS) finite element discretization [1]. This hybrid continuous-discontinuous Petrov-Galerkin method uses solution (trial) functions that are piecewise continuous over the whole domain. That is, these functions correspond to standard finite element partitions. We then use as weight (test) functions piecewise discontinuous bases. These broken test spaces allow us to extend the DPG approach [2] to compute optimal test functions automatically and apply these to establish numerically stable FE approximations. Important features of this discretization are as follows. The support of each discontinuous test function is identical to its corresponding continuous trial function. The local test-function contribution is computed locally on an element by element fashion (i.e. decoupled). This has a linear cost with respect to the problem size and can be thought as an alternative assembly process, where not only inner products, but the functions themselves need to be computed on the fly. Additionally, our experience indicates that the computation of the optimal test functions is achieved with sufficient accuracy by using the same polynomial order of approximation as used in the trial function. As in every other DPG formulation, the resulting algebraic system is symmetric and positive definite, allowing us to use simple iterative strategies to compute the numerical solution. Our future work will include developing variationally stable discretizations based on isogeometric analysis (IGA) both in Galerkin as well as in collocation form. Our preliminary results indicate that these methods are very promising by delivering robust and efficient discretizations exploiting the smoothness of IGA basis functions to deliver intrinsically stable discretizations that are symmetric and positive definite for arbitrary partial differential equations.

REFERENCES [1] V. M. Calo, A. Romkes, and E. Valseth, Automatic Variationally Stable Analysis for FE Computations: An Introduction, Lecture Notes in Computational Science and Engineering, Submitted, http://arxiv.org/abs/1808.01888 [2] L. F. Demkowicz and J. Gopalakrishnan, Analysis of the DPG Method for the Poisson Equation, SIAM Journal on Numerical Analysis, 49(5), 1788-1809, 2011.



SUM FACTORIZATION TECHNIQUES IN IGA

A. Bressan1, S. Takacs2

1 University of Oslo [email protected]

2 RICAM, Austrian Academy of Science, Linz [email protected]

ABSTRACT

The fast assembling of stiffness and mass matrices is a key issue in IGA, particularly for high

degree splines. If the assembling is done as in FEM, the computational complexity grows with the

spline degree to a power 3d, where d is the domain dimension. Assembling procedures with

reduced complexity were recently developed. Examples are sum factorization [1], low rank

assembling [2,4], and weighted quadrature [3]. In [1] it was shown that the computational

complexity of the sum factorization approach grows with the spline degree to a power of 2d+1.

We show that it is possible to decrease the power to d+2 without loosing generality or accuracy.

REFERENCES

[1] P. Antolin, A. Buffa, F. Calabrò, M. Martinelli, and G. Sangalli, Efficient matrix computation

for tensor-product isogeometric analysis: The use of sum factorization, Computer Methods in

Applied Mechanics and Engineering, 285, 817 – 828, 2015.

[2] A. Mantzaflaris, B. Jüttler, B. N. Khoromskij, and U. Langer, Low rank tensor methods in

Galerkin-based isogeometric analysis, Computer Methods in Applied Mechanics and

Engineering Special Issue on Isogeometric Analysis: Progress and Challenges, 316, 1062 –

1085, 2017.

[3] F. Calabrò, G. Sangalli, and M. Tani, Fast formation of isogeometric Galerkin matrices by

weighted quadrature, Computer Methods in Applied Mechanics and Engineering Special Issue

on Isogeometric Analysis: Progress and Challenges, 316, 606 – 622, 2017.

[4] C. Hofreither, A black-box low-rank approximation algorithm for fast matrix assembly in

isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 333, 311 – 330,

2018.

isogeometric analysis for 2d and 3d mhd subproblems...

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