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Lect09-p2 - March17,2010 StanTomov CS594 Slide1/19 Outline...

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  Slide 1 / 19 Mesh Generation and Load Balancing CS 594 03/17/2010 Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee March 17, 2010
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  Slide 2 / 19 Outline Motivation Reliable & efficient PDE simulations for high end computing systems Background PDE simulation concept: approximation is over a mesh Error Analysis Simulation error: related to “local mesh size” Adaptive Mesh Generation Support parallel refinement/derefinement and “element migration” Load Balancing Scalability of the computation on modern architectures Data structures Algorithmically motivated: multigrid, domain decomposition, etc. For performance optimization: architecture aware computing Numerical Example  Conclusions
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  Slide 3 / 19 Motivation PDE simulations have  errors   stemming from the  numerical approximation (related to the mesh, ...) The need for Reliable “error” to be less than desirable tolerance       and Efficient do not do “overkill” computation       PDE simulations for  High end computing systems .
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  Slide 4 / 19 Background In general: “Error” from the discretization is proportional to the mesh size A problem:  localized  physical phenomena deteriorate the approximation  properties of classical PDE approximations How can we find a “good” mesh, i.e. yielding  small and reliable error   and  efficient computation For example flows near wells; faults; moving fronts, etc.
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  Slide 5 / 19 Background Solution:   (1) determine (automatically) the regions of singular                          behaviour, and                    (2) refine them in a “balanced” manner Example: Efficiency of locally adapted vs uniform approximation of on an L-shaped domain r 1 / 2 sin θ / 2
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  Slide 6 / 19 Background Computational framework of the  Adaptive methods: i.e. a process of continuous feedback from the computation to find a reliable and efficient numerical PDE approximation Solve PDE Evaluate the approximation “error” Is “error”  acceptable Improve the approximation (h/p refinement) no yes done
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  Slide 7 / 19 Error Analysis The numerical solution of PDE (e.g. FEM) Boundary value problem:       Au      =  f ,    subject to boundary conditions Get a “weak” formulation:   (Au,  ϕ
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