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- 1. Spacetime Meshing for Discontinuous Galerkin Methods Shripad Thite Ph.D. defense — August 22, 2005
- 2. Collaborators Members of the Center for Process Simulation & Design (CPSD) CS Jonathan Booth, Sayantan Chakraborty, Shuo-Heng Chung, Prof. Jeﬀ Erickson, Prof. Michael Garland, Harigovind Gopinatha- panicker, Damrong Guoy (also CSAR), Mark Hills, Prof. Sanjay Kale, Shripad Thite, Terry Wilmarth, Yuan Zhou Math Prof. John Sullivan (also TU-Berlin) TAM Reza Abedi, Yong Fan, Prof. Robert Haber, Morgan Hawker, Jayandran Palaniappan Supported by NSF ITR grant DMR-0121695 1
- 3. Publications Spacetime Meshing with Adaptive Reﬁnement and Coarsening Reza Abedi, Shuo-Heng Chung, Jeﬀ Erickson, Yong Fan, Michael Garland, Damrong Guoy, Robert Haber, John M. Sullivan, Shripad Thite, Yuan Zhou 20th Symp. Computational Geometry, June 2004 An h-adaptive Spacetime-Discontinuous Galerkin Method for Linearized Elastody- namics Reza Abedi, Robert Haber, Shripad Thite, Jeﬀ Erickson ´e Submitted to Revue Europ´enne des El´ments Finis, 2005 e Eﬃcient Spacetime Meshing with Nonlocal Cone Constraints Shripad Thite 13th International Meshing Roundtable, September 2004 A Uniﬁed Algorithm for Adaptive Spacetime Meshing with Nonlocal Cone Constraints Shripad Thite 21st European Workshop on Computational Geometry, March 2005 Invited to special issue of Computational Geometry: Theory and Applications (CGTA) 2
- 4. Motivation Important applications in science and engineering involve simu- lating transient time-dependent phenomena e.g., Conservation Laws in Elastodynamics and Fluid Dynamics. Ocean waves, Acoustics, Gas dynamics, Traﬃc ﬂow Computer simulation of such phenomena involves solving space- time hyperbolic PDEs e.g., Wave traveling along a taut string with wavespeed ω: utt − ω 2 uxx = 0 3
- 5. Linear elastodynamics σ σ When a rectangular plate with a crack in the center is loaded, the shock wave scatters oﬀ the crack-tip Meshing: Shuo-Heng Chung, Shripad Thite; Solution: Reza Abedi; Visualization: Yuan Zhou 4
- 6. My thesis topic A spacetime mesh is a partition of the spacetime domain into simplices (triangles, tetrahedra, etc.) Given a spacetime mesh Ω, Spacetime Discontinuous Galerkin (SDG) methods compute the approximate numerical solution over Ω I give algorithms to generate spacetime meshes in dD×time The algorithms in my thesis support an eﬃcient, parallelizable, O(N )-time solution strategy by SDG methods I prove worst-case guarantees on the size and quality of the mesh 5
- 7. Causality 6
- 8. Cone of inﬂuence Point A inﬂuences point B (A B) iﬀ changing the parameters at A could possibly change the solution at B Points inﬂuenced by A are approximated by a cone of inﬂuence with apex A and slope 1/ω(A) B t y A x If neither A B nor B A, then A and B are independent 7
- 9. Coupling Spacetime element A inﬂuences spacetime element B (A B) iﬀ some point of A inﬂuences some point of B If A B, then A must be solved no later than B If A B and B A; then, A and B are coupled and must be solved simultaneously A B 8
- 10. Patches A patch Π is a set of elements that must be solved simultaneously A patch plus its inﬂow information is a solvable unit B A We generate meshes incrementally patch-by-patch so that Total computation time = patch Π∈Ω Time to solve Π is bounded 9
- 11. Advancing Front Meshing 10
- 12. Basic incremental algorithm Input: triangulation of the d-dimensional space domain M at time t = 0 Output: (d + 1)-dimensional spacetime mesh Ω of M × [0, ∞) Construct a sequence of fronts τ0, τ1 , τ2, . . ., embedded in spacetime—create small patches between successive fronts A front is a terrain t front space mesh x 11
- 13. Basic incremental algorithm Initialize τ0 ← Mt=0 For i = 0, 1, 2, ... Advance a local neighborhood N of τi to N Triangulate the volume between τi and τi+1 Solve the resulting patch Each new front is obtained from the previous front by a local operation 12
- 14. Every front must be causal A front is causal iﬀ it is an independent set—no two points of a front inﬂuence each other The slope of a causal front is less than that of the cone of inﬂuence of each of its points time A 13
- 15. Tent Pitcher Advances in time a vertex P to P Greedily maximizes the height of the tentpole P P subject to causality outflow faces new front tentpole old front implicit faces inflow faces Old and new fronts causal =⇒ patch can be solved immediately 14
- 16. Tent Pitcher plus In 2D×time and higher dimensions, being too greedy at each ¨o step can prevent progress in future [Ung¨r and Sheﬀer, IMR’00] Erickson et al. [Engg. with Computers, ’05] devised progress constraints Causality and progress constraints guarantee progress, propor- tional to local geometry, when a local minimum vertex is pitched 15
- 17. My improvements to Tent Pitcher 1. Adapt mesh resolution to error estimates in 2D×time 2. Adapt duration of spacetime elements to changing wavespeeds in dD×time All this requires new constraints and a new meshing algorithm which I derive and prove correct 16
- 18. Improvements to Tent Pitcher (cont’d.) 3. Adapt size and position of mesh features to track moving boundaries I propose a set of more general front advancing operations re- quired and useful for boundary tracking I give a framework to perform various operations when possible and desirable 17
- 19. Technical Details 18
- 20. Progress constraint Limit progress at each step to guarantee progress in the next step R 0 p q lowest lowest 1 r P Q p q [Damrong Guoy] 19
- 21. Next step is legal R 0 lift p 1 lift q r 2 P Q p q 20
- 22. Causality violated in next step R 0 1 lift p 3 r lift q Q p q 4 21
- 23. Forbidden zones R gradient of qr gradient of pr r Q p q Progress constraint 1/ω limits slope of QR and of P R 22
- 24. Adapting to changing wavespeeds Wavespeeds can be solution-dependent for nonlinear PDEs Increasing wavespeed breaks old algorithm At every step, the front must satisfy a progress constraint that anticipates the wavespeed in the next step I derive a progressive invariant that guarantees progress even when wavespeed increases discontinuously I give an algorithm to greedily maximize progress subject to this invariant 23
- 25. No focusing: ω(P ) ≤ maxP ∈cone(Q){ω(Q)} P P Q Q Allows us to conservatively estimate future wavespeed given the cone of inﬂuence everywhere on the current front 24
- 26. Handling faster wavespeeds When wavespeed is not constant, we cannot be greedy because the amount of progress in the current step depends on the future wavespeed I give an algorithm to look ahead h tent pitching steps to con- servatively estimate future wavespeed The new algorithm maximizes tentpole height in the current step subject to this conservative estimate 25
- 27. Look ahead one step P QR < 1/ω(P QR) 1/ω(P QR) 0 1 r Q P p q 26
- 28. Look ahead one step R 0 1/ω(P QR) 1/ω(P QR) 2 1 r P QR = 1/ω(P QR) Q P p q 27
- 29. Look ahead one step R 0 1/ω(P QR) 1/ω(P QR) 1/ωmax 3 2 1 r Q P p q 28
- 30. Invariant: h-progressive Let δ > 0 be a function of the shape of pqr. Triangle P QR is h-progressive iﬀ it is causal and h = 0: P QR is 0-progressive if it satisﬁes progress con- straint 1/ωmax h > 0: P QR is h-progressive if after pitching local mini- mum P of P QR to P by δ, P QR is (h − 1)-progressive Horizon h limits number of lookahead steps 29
- 31. Nonlocal cone constraints When wavespeed is not constant, most limiting cone constraint can be nonlocal 30
- 32. Reﬁnement and coarsening B C A D D B C C B A A B C B C Newest vertex bisection [Sewell ’72, Mitchell ’88] Spacetime mesh is not conforming! Bisecting triangles on the current front decreases size of future spacetime elements Coarsening ≡ De-reﬁnement 31
- 33. Adaptive progress constraint D A B C C B A D Gradient vector is excluded from eight forbidden zones 32
- 34. Linear adaptive example 33
- 35. Mesh adaptivity with nonlinearity I give an algorithm to adapt size (to error estimates) and duration (to changing wavespeed) of elements Unify (i) adaptive progress constraint, and (ii) lookahead algo- rithm for meshing with nonlocal cone constraints During lookahead, if the algorithm predicts a bad quality element due to increasing wavespeed, then it can preemptively reﬁne the front Reﬁnement improves or maintains temporal aspect ratio More progress can be made in the current step if the algorithm prepares for future reﬁnement 34
- 36. New invariant: (h, l)-progressive A front τ is (h, l)-progressive if and only if it is causal and 1. τ is h-progressive; and 2. bisect(τ ) is (h, max{l − 1, 0})-progressive. Base case l = 0: τ is (h, 0)-progressive if bisectk (τ ) is h-progressive for every k ≥ 0. Progress constraint adapts to the level of reﬁnement l 35
- 37. Example I 36
- 38. Example II 37
- 39. Main result Given a simplicial mesh M ∈ Ed , our algorithm builds a simplicial mesh Ω of the spacetime domain M × [0, ∞) that satisﬁes all the following criteria: For every T ≥ 0 the spacetime volume M × [0, T ] is contained in the union of a ﬁnite number of simplices of Ω The minimum temporal aspect ratio of any spacetime element is bounded from below tentpole duration duration t height x 38
- 40. Main result (cont’d.) Additionally, in 2D×Time, Our algorithm adapts the size of spacetime elements to a pos- teriorierror estimate Provided the number of reﬁnements is ﬁnite, our algorithm ter- minates with a ﬁnite mesh of M × [0, T ] for every target time T ≥0 Other results omitted from this talk 39
- 41. New meshing operations r r s s m p p q q Inclined tentpoles Edge bisection S Q P R s p q r Edge ﬂips Vertex deletion 40
- 42. Boundary tracking Geometry and topology of domain changes over time e.g., combustion of solid rocket fuel The mesh must adapt by changing the size of mesh elements, or by varying the placement of mesh features, or both We incorporate techniques like smoothing, usually performed as a global remeshing step, into our local advancing framework 41
- 43. Tracking simple motion I devised a policy to assign priorities to various front advancing operations: 1. Coarsen the front by deleting a vertex u 2. Flip an edge of the front if it improves the spatial aspect ratio 3. Pitch an interior vertex to smooth local triangulation 4. Pitch a boundary vertex in prescribed direction 42
- 44. Additional heuristics If an obtuse angle gets too big, ﬂip or bisect the opposite edge Bisect an edge if its endpoints have very diﬀerent velocities; choose an average velocity for new midpoint Each operation performed only when allowed by the solver 43
- 45. An example 44
- 46. Conclusion 45
- 47. Summary of results Adapt mesh resolution to a posteriori numerical error estimate Spacetime Meshing with Adaptive Reﬁnement and Coarsening Reza Abedi, Shuo-Heng Chung, Jeﬀ Erickson, Yong Fan, Michael Garland, Damrong Guoy, Robert Haber, John M. Sullivan, Shripad Thite, Yuan Zhou 20th Symp. Computational Geometry, June 2004 An h-adaptive Spacetime-Discontinuous Galerkin Method for Linearized Elastody- namics Reza Abedi, Robert Haber, Shripad Thite, Jeﬀ Erickson ´e Submitted to Revue Europ´enne des El´ments Finis, 2005 e 46
- 48. Summary of results (cont’d.) Adapt to changing wavespeeds Eﬃcient Spacetime Meshing with Nonlocal Cone Constraints Shripad Thite 13th International Meshing Roundtable, September 2004 A Uniﬁed Algorithm for Adaptive Spacetime Meshing with Nonlocal Cone Constraints Shripad Thite 21st European Workshop on Computational Geometry, March 2005 Invited to special issue of Computational Geometry: Theory and Applications (CGTA) 47
- 49. Summary of results (cont’d.) Give more general meshing operations in spacetime Propose a framework for smoothing to improve mesh quality, and for tracking moving boundaries and other singular surfaces 48
- 50. Impact at a glance My thesis extends Tent Pitcher to eﬃciently solve more general problems Nonadaptive linear Nonadaptive nonlinear Adaptive linear Adaptive nonlinear Produce an eﬃciently solvable non-degenerate mesh in all these cases Smoothing and boundary tracking do not create inverted ele- ments If the front triangles degrade in spatial quality more than ex- pected, the algorithm can get stuck 49
- 51. Some future directions Extend algorithms to arbitrary dimensions, ﬁrst 3D×time e.g. Devise an adaptive meshing algorithm in 3D×time Give a provably correct and complete boundary tracking algo- rithm for interesting classes of motion Handle changes in topology of domain over time 50
- 52. Thank you! 51

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