Fully automatic hp-adaptivity for acoustic and electromagnetic scattering in three dimensions

textWe present an algorithm for fully automatic hp-adaptivity for finite element approximations of elliptic and Maxwell boundary value problems in three dimensions. The algorithm automatically generates a sequence of coarse grids, and a corresponding sequence of fine grids, such that the energy norm of error decreases exponentially with respect to the number of degrees of freedom in either sequence. At each step, we employ a discrete optimization algorithm to determine the refinements for the current coarse grid such that the projection-based interpolation error for the current fine grid solution decreases with an optimal rate with respect to the number of degrees of freedom added by the refinement. The refinements are restricted only by the requirement that the resulting mesh is at most 1-irregular, but they may be anisotropic in both element size h and order of approximation p. While we cannot prove that our method converges at all, we present numerical evidence of exponential convergence for a diverse suite of model problems from acoustic and electromagnetic scattering. In particular we show that our method is well suited to the automatic resolution of exterior problems truncated by the introduction of a perfectly matched layer. To enable and accelerate the solution of these problems on commodity hardware, we include a detailed account of three critical aspects of our implementations, namely an efficient implementations of sum factorization, several interfaces to the direct multi-frontal solver MUMPS, and some fast direct solvers for the computation of a sequence of nested projections.Computational Science, Engineering, and MathematicsComputational and Applied Mathematic

Multi-dimensional Limiting Strategy for Higher-order CFD Methods - Progress and Issue (Invited)

The present paper deals with the progress of multi-dimensional limiting process (MLP) and discuss the issues for further improvements. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillationcontrol mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn approximation or reconstruction. The resulting algorithm, called the hierarchical MLP, facilitates the capturing of detailed flow structures while maintaining the formal order-of-accuracy in smooth region and providing accurate non-oscillatory solutions across discontinuous region. This algorithm has been developed within the modal DG framework, but it also can be formulated into a nodal framework, most notably the CPR framework. Troubled-cells are detected by applying the MLP concept, and the final accuracy is determined by the projection procedure and the hierarchical MLP limiting step. Through extensive numerical analyses and computations ranging from scalar conservation laws to fluid systems, it is demonstrated that the proposed limiting approach yields the outstanding performances in capturing compressible inviscid and viscous flow features. Further issues are also mentioned to improve and extend the current approach for higher-order simulations of high-Reynolds number compressible flows.Authors appreciate the financial supports by the EDISON program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2011-0020559) and by NSL (National Space Laboratory) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2014M1A3A3A02034856). This work is also partially supported by the RoK ST&R project of Lockheed Martin Corporation. Authors also appreciate the computing resources provided by the KISTI Supercomputing Center(KSC-2014-C3-054).OAIID:RECH_ACHV_DSTSH_NO:420150000004648007RECH_ACHV_FG:RR00200003ADJUST_YN:EMP_ID:A001138CITE_RATE:FILENAME:6.2015-3199.pdfDEPT_NM:기계항공공학부EMAIL:[email protected]_YN:FILEURL:https://srnd.snu.ac.kr/eXrepEIR/fws/file/a984d649-4b23-435b-adc9-df9aa0c8aa46/linkCONFIRM:

Discontinuous Galerkin Methods for Solving Acoustic Problems

Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.Partial differential equations play an important role in engineering applications. It is often possible to solve these equations only approximately, i.e. numerically. Therefore number of successful discretization techniques has been developed to solve these equations. The presented discontinuous Galerkin method seems to be very general method to solve this type of equations, especially useful for hyperbolic systems. Our aim is to solve aeroacoustic problems, where propagation of acoustic waves is described using linearized Euler equations. This system of equations is indeed hyperbolic and therefore the discontinuous Galerkin method was chosen. The most important aspects of this method is ability to deal with complex geometries, possibility of high-order method and its local character enabling efficient computation parallelization. We first introduce the discontinuous Galerkin method in general for one- and two-dimensional problems. We then test the algorithm to solve advection equation, which was chosen as a model case of hyperbolic equation. The method will be finally tested using number of verification problems, which were formulated to test methods for computational equations, including verification of boundary conditions, which, similarly to computational fluid dynamics, are important part of computational aeroacoustics.

Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

AIAA SciTech, 52nd Aerospace Sciences Meeting 13-17 January 2014, National Harbor, MarylandHigher-order multi-dimensional limiting Process (MLP) [J. S. Park and C. Kim, Higher-order Multi-dimensional Limiting Strategy for Discontinuous Galerkin Methods in Compressible Inviscid and Viscous Flows, Comp. & Fluids, In press] is improved and applied to correction procedure via reconstruction (CPR) on unstructured grids. MLP, which has been originally developed in nite volume method (FVM), provides an accurate, robust and ecient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn reconstruction. The resulting algorithms, called the hierarchical MLP, facilitate the accurate capturing of detailed flow structures in both continuous and discontinuous regions. This algorithm has been developed in the modal DG framework, but it also can be formulated into a nodal framework, most notably the CPR framework. Troubled-cells are detected by applying the MLP concept, and the nal accuracy is determined by the projection procedure and MLP limiting step. Through extensive numerical analyses and computations, it is demonstrated that the proposed limiting approach yields the desired accuracy and outstanding performances in resolving compressible inviscid and viscous flow features.This work is supported by NSL (National Space Lab.) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant 20090091724), by the third stage of the Brain Korea 21 Plus Project in 2013, and by Korea Ministry of Land, Transport and Maritime A airs as Haneul Project. This work is also supported by the KISTI Supercomputing Center (KSC-2012-C3-40) and by PLSI supercomputing resources of KISTI.OAIID:oai:osos.snu.ac.kr:snu2014-01/104/0000004648/1SEQ:1PERF_CD:SNU2014-01EVAL_ITEM_CD:104USER_ID:0000004648ADJUST_YN:NEMP_ID:A001138DEPT_CD:446CITE_RATE:0FILENAME:6%2e2014-0772.pdfDEPT_NM:기계항공공학부EMAIL:[email protected]:

An SEM-DSM three-dimensional hybrid method for modelling teleseismic waves with complicated source-side structures

Despite recent advances in High Performance Computing (HPC), numerical simulation of high frequency (e.g. 1 Hz or higher) seismic wave propagation at the global scale is still prohibitive. To overcome this difficulty, we propose a hybrid method to efficiently compute teleseismic waveforms with 3-D source-side structures. By coupling the Spectral Element Method (SEM) with the Direct Solution Method (DSM) based on the representation theorem, we are able to limit the costly SEM simulation to a small source-side region and avoid computation over the entire space of the Earth. Our hybrid method is benchmarked against 1-D DSM synthetics and 3-D SEM synthetics. We also discuss numerical difficulties in the implementation, including slow DSM convergence near source depth, discretization error, Green’s function interpolation and local 3-D wavefield approximations. As a case study, we apply our hybrid method to two subduction earthquakes and show its advantage in understanding 3-D source-side effects on teleseismic P-waves. Our hybrid method reduces computational cost by more than two orders of magnitude when only source-side 3-D complexities are of concern. Thus our hybrid method is useful for a series of problems in seismology, such as imaging 3-D structures of a subducting slab or a mid-ocean ridge and studying source parameters with 3-D source-side complexities using teleseismic waveforms

Algorithmische und Code-Optimierungen Molekulardynamiksimulationen für Verfahrenstechnik

The focus of this work lies on implementational improvements and, in particular, node-level performance optimization of the simulation software ls1-mardyn. Through data structure improvements, SIMD vectorization and, especially, OpenMP parallelization, the world’s first simulation of 2*1013 molecules at over 1 PFLOP/sec was enabled. To allow for long-range interactions, the Fast Multipole Method was introduced to ls1-mardyn. The algorithm was optimized for sequential, shared-memory, and distributed-memory execution on up to 32,768 MPI processes.Der Fokus dieser Arbeit liegt auf Code-Optimierungen und insbesondere Leistungsoptimierung auf Knoten-Ebene für die Simulationssoftware ls1-mardyn. Durch verbesserte Datenstrukturen, SIMD-Vektorisierung und vor allem OpenMP-Parallelisierung wurde die weltweit erste Petaflop-Simulation von 2*1013 Molekülen ermöglicht. Zur Simulation von langreichweitigen Wechselwirkungen wurde die Fast-Multipole-Methode in ls1-mardyn eingeführt. Sequenzielle, Shared- und Distributed-Memory-Optimierungen wurden angewandt und erlaubten eine Ausführung auf bis zu 32768 MPI-Prozessen

A study on block flexible iterative solvers with applications to Earth imaging problem in geophysics

Les travaux de ce doctorat concernent le développement de méthodes itératives pour la résolution de systèmes linéaires creux de grande taille comportant de nombreux seconds membres. L’application visée est la résolution d’un problème inverse en géophysique visant à reconstruire la vitesse de propagation des ondes dans le sous-sol terrestre. Lorsque de nombreuses sources émettrices sont utilisées, ce problème inverse nécessite la résolution de systèmes linéaires complexes non symétriques non hermitiens comportant des milliers de seconds membres. Dans le cas tridimensionnel ces systèmes linéaires sont reconnus comme difficiles à résoudre plus particulièrement lorsque des fréquences élevées sont considérées. Le principal objectif de cette thèse est donc d’étendre les développements existants concernant les méthodes de Krylov par bloc. Nous étudions plus particulièrement les techniques de déflation dans le cas multiples seconds membres et recyclage de sous-espace dans le cas simple second membre. Des gains substantiels sont obtenus en terme de temps de calcul par rapport aux méthodes existantes sur des applications réalistes dans un environnement parallèle distribué. ABSTRACT : This PhD thesis concerns the development of flexible Krylov subspace iterative solvers for the solution of large sparse linear systems of equations with multiple right-hand sides. Our target application is the solution of the acoustic full waveform inversion problem in geophysics associated with the phenomena of wave propagation through an heterogeneous model simulating the subsurface of Earth. When multiple wave sources are being used, this problem gives raise to large sparse complex non-Hermitian and nonsymmetric linear systems with thousands of right-hand sides. Specially in the three-dimensional case and at high frequencies, this problem is known to be difficult. The purpose of this thesis is to develop a flexible block Krylov iterative method which extends and improves techniques already available in the current literature to the multiple right-hand sides scenario. We exploit the relations between each right-hand side to accelerate the convergence of the overall iterative method. We study both block deflation and single right-hand side subspace recycling techniques obtaining substantial gains in terms of computational time when compared to other strategies published in the literature, on realistic applications performed in a parallel environment