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UW-Madison ME 964 - Outlining Midterm Projects

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ME964High Performance Computing for Engineering Applications“The real problem is not whether machines think but whether men do.”B. F. Skinner© Dan Negrut, 2011ME964 UW-MadisonOutlining Midterm ProjectsTopic 3: GPU-based FEATopic 4: GPU Direct Solver for Sparse Linear AlgebraMarch 01, 2011Before We Get Started… Last time Midterm Project topics 1 and 2 Discrete Element Method on the GPU. Area coordinator: Toby Heyn Collision Detection on the GPU. Area coordinator: Arman Pazouki Today Midterm Project topics 3 and 4 Finite Element Method on the GPU. Area coordinators: Prof. Suresh and Naresh Khude Sparse direct solver on the GPU (Cholesky). Area coordinator: Dan Negrut Midterm Project Related Issues Midterm Project is due on 04/13 at 11:59 PM (use Learn@UW drop-box) Intermediate report due on 03/22 at 11:59 PM (use the same Learn@UW drop-box) Each area coordinator Will provide a test problem for you to test your GPU implementation Will also assist you with questions related to the non-programming aspects (the “theory”) behind the topic you chose You can continue your Midterm Project (MP) and have it become your Final Project (FP) In this case you will be expected to show how the FP implementation is superior to your MP implementation Other issues HW5 due tonight at 11:59 PM Use Learn@UW drop-box to submit homework2Finite Element Analysison the GPU?Krishnan [email protected] ProfessorFinite Element Analysis Computer simulation of engineering models Physics:– Structural, thermal, fluid, … Mode:– Static, modal, transient– Linear, non-linear, multi-physicsWhy GPU?Hours or even days of CPU time.[Gordon; JPL]QuestionCan one exploit graphics programmable units (GPU) to speed-up Finite Element analysis?+Structural Static FEAModel DiscretizePost-processElementStiffnesseeKfAssemble/SolveKu f=eeK Kf f==∑∑FEA: VariationsDiscretizeModelElementStiffnessAssemble/SolvePost-processeeK Kf f==∑∑Ku f=NonlinearOptimizationTet/Hex/…Direct/IterativeOrder/HybrideeKfFEA: ChallengesDiscretizeModelElementStiffnessAssemble/SolvePost-processeeK Kf f==∑∑Ku f=NonlinearOptimizationTet/Hex/…Direct/IterativeOrder/HybrideeKf1. Accuracy2. Automation3. SpeedTypical BottleneckModel DiscretizePost-processElementStiffnesseeKfAssemble/SolveKu f=eeK Kf f==∑∑GPU & Engineering AnalysisModel DiscretizeCPUGPU?Discretization Data: Small b-rep (+) Logic: Complex (-) Threads: Few (-)Not a good candidate for GPU!?Element StiffnessElement Stiffness Data: O(N) (+/-) Logic: Simple (+) Threads: N (+)DiscretizeModelElementStiffnesseeKfCPUCPUGPU?Hex 2ndOrder Hex HybridStiffness: Hex 2ndOrder[]( , )eM MK = 8 Corners~100 Bytes Data (x y z) 27 Nodes~ M = 81 DOF (u v w) kij~ Gaussian integration– 30 flops(8 Corners)(27 Nodes)2(15 )Flops N M≈200000, 814secCPUN MT= =≈Typical BottleneckModel DiscretizeElementStiffnesseeKfAssemble/SolveKu f=eeK Kf f==∑∑Direct vs. IterativeKu f=K is sparse & usually symmetric P.D1 1TTK LDLu L D L f− − −==Direct1( ): Preconditioner of Ki i iu u B f KuB+= + −Iterative(GPU Variation: Assembly-free)Note: Nvidia offers CuBLAS-3 dense matrix libraryDirect Sparse on GPU (1)(2006)Direct Sparse on GPU (1)Ku f=Direct Sparse on GPU (1)Ku f=Direct Sparse on GPU (2)Ku f=(2008)Direct Sparse on GPU (2)Ku f=Iterative Sparse on GPU (1)(2008) Jacobi preconditioned conjugate gradient ATI GPU Speed-up 3.5.Iterative Sparse on GPU (2) Double precision real world SpMv – CPU (2.3 GHz Dual Xeon): 1 GFLOPS– GPU (GTX 280): 16 GFLOPS– Speedup ~ 16FEA/GPU Class Projects?1. Complete < 6 weeks2. Important (publishable)3. Pilot codeFEA/GPU Class Projects?1. GPU Friendly Preconditioners for Thin Structures– Research papers– OpenCL and ViennaCL Pilot Code2. Topology Optimization – Research papers– CUDA code3. Others – Can discuss …Thin Structure?Thin Structure?Large KPreconditioners?Ku f= Iterative Methods: – GPU methods available for K*u– Typical preconditioners: simple Jacobi, … Poor preconditioner … slow convergence Objective:– GPU friendly preconditioner for thin structures1( ): Preconditioner of Ki i iu u B f KuB+= + −Research PublicationBasic IdeaAlgorithmWhy Preconditioner?Why Double Precision?How Expensive is Preconditioner?GPU FriendlySpeed-up without PreconditionerSpeed-up with PreconditionerFEA/GPU Class Projects?1. GPU Friendly Preconditioners for Thin Structures– Research papers– OpenCL and ViennaCL Pilot Code2. Topology Optimization – Research papers– CUDA code3. Others – Can discuss …Topology Optimization0 JDMinVΩ⊂Ω =0 {J, }DMin VΩ⊂D[Sigmund 2001] V = 50%Stiffest topology for a given volume?Where to remove material?Multi Objective + Topology Optimization = MOTODemoMatlab code www.ersl.wisc.eduPareto Optimal Designs Purely pareto optimalComparisonD3-DPareto-MethodSIMP3-D GPU ImplementationMulti-grid Topology Optimization on the GPU (IDETC conf. 2011)Motivation for Topic 4:Sparse Direct Solver42Nomenclature& Simplifying Assumptions43The Schur Complement Problem inMulti-Body Dynamics Applications44Formulation Framework Position:  Orientation: Euler parameters, Translational Velocity:  Angular velocities , , ]x y y Ti i i iω = [ω ω ωω = [ω ω ωω = [ω ω ωω = [ω ω ω0 1 2 3[ , , , ]Ti i i i ie e e e=p[ , , ]Ti i i ix y z=rɺɺ ɺɺ[ , , ]Ti i i ix y z=r45Constrained Equations of Motion( , , )tΦΦΦΦ=r p 0( , , ) ( , , ) ( , , )tt t tη ρη ρη ρη ρΦ Φ ω ΦΦ Φ ω ΦΦ Φ ω ΦΦ Φ ω Φ+ = −r p r r p r pɺ( , , ) ( , , ) ( , , , , )t t tη ρη ρη ρη ρΦ Φ ω τ ωΦ Φ ω τ ωΦ Φ ω τ ωΦ Φ ω τ ω+ =r p r r p r r pɺɺɺ ɺ( , , )( , , , , )ˆ( , , )( , , , , )TTttttηηηηρρρρΦΦΦΦ ωωωωλλλλΦΦΦΦω ωω ωω ωω ω      + =            r pM 0 r F r r pr p0 J n r r pɺɺ ɺɺɺ46Numerical Solution of the Newton-Euler Constrained Equations of Motion One has to solve a set of Differential Algebraic Equations (DAEs) to find the time evolution of a mechanical system Most often the numerical solution of the DAEs requires the solution of a linear system of the form:ˆTTηηηηρρρρη ρη ρη ρη ρΦΦΦΦΦ ωΦ ωΦ ωΦ ωΦ Φ λ τΦ Φ λ τΦ Φ λ τΦ Φ λ τ        =    


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