Representing Arbitrary Bounding Surfaces in the Material Point Method Carter M. Mast 6 th MPM Workshop Albuquerque, New Mexico August 9-10, 2010 Participants: Peter Mackenzie-Helnwein, Pedro Arduino, and Greg Miller Department of Civil and Environmental Engineering University of Washington Seattle, WA
Outline Motivation and Overview Approach Implementation Outlook/Future Research 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 2
Motivation Loading on structures due to landslide/debris flows Landslide/Debris flow Appropriate numerical method (MPM) Material models Phase transition Etcetera Domain Topological l --- e.g. hll hillside Structural Both require general surfaces 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 3
Motivation Loading on structures due to landslide/debris flow Landslide/Debris flow Appropriate numerical method (MPM) Material models Phase transition Etcetera Domain Topological l --- e.g. hll hillside Structural Both require general surfaces 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 4
Overview Disadvantages of this surface representation: Surface is dependent on the computational nodes 8/9/2010 Unrealistic Carter Mast - University of Washington - 6th MPM Workshop 5
Overview Potential solutions/fix to the surface representation problem: Refine the mesh Represent esent the surface as a rigid body Irregular mesh over entire domain Computational tion expensive e Increase total number of particles Search algorithm Meshing algorithm 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 6
Overview Potential solutions to the surface representation problem: Refine the mesh Represent esent the surface as a rigid body Irregular mesh over entire domain Computational tion expensive e Increase total number of particles Search algorithm Meshing algorithm Introduce a second grid Dual-Grid Approach 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 7
Approach Ω A Ω α Dual-Grid methodology Introduce a separate (additional) grid that follows the geometry of the bounding surface Two grids, one body Effectively communicate dynamic information between the two grids 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 8
Approach Dual-Grid methodology The Blending Approach Each grid is used to create independent fields for velocity and acceleration Piecewise description: 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 9
Approach Ω A Ω α Φ The Blending Approach Enforce continuity along Φ Leads to the constraint of the form 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 10
Approach The Blending Approach Algorithmic implementation: 1. Use the traditional MPM algorithm to solve for nodal acceleration and velocity at time t n for those nodes in the boundary grid. 2. Solve for the nodal accelerations on the standard grid. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 11
Approach The Blending Approach Algorithmic implementation: 3. Solve for the nodal velocities on the standard grid. 4. Update nodal values for both grids. 5. Update particles: a. For particles with p then the update comes from boundary grid nodes. b. For particles with then the update comes form p standard d grid nodes. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 12
Implementation Evaluate algorithm using one- dimensional test case Uniaxial steel bar subjected to rigid boundary Standard d MPM: 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 13
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Implementation Arbitrary boundary representation in 1-d: 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 16
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Implementation From the 1-d results: Certain configurations (boundary location and boundary cell size) are problematic. Particularly for the Enhanced approach. The Blending approach provides more consistent results Boundary grid size should be similar to the standard grid size. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 21
Implementation From the 1-d results: Certain configurations (boundary location and boundary cell size) are problematic. Boundary grid size should be similar to the standard grid size. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 22
Implementation Moving into 2- and 3-d Increasing complexity Boundary orientation Evaluation of the integral linking the two grids Fully 3-d code restricted to planar boundaries with regular node spacing 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 23
Implementation A relatively straight forward problem 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 24
Implementation Single particle analysis 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 25
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Implementation Single particle analysis Discretization A 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 32
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Implementation Single particle analysis Discretization A Discretization B 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 39
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Outlook Successfully models all boundary locations/orientations for the single particle The standard d MPM results are recovered Very little consistency from one boundary location to another Body or grid discretization error? Formulation error? Coding error? 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 46
Future Work Continue to try and find out why/what is causing the inconsistency. Implement the alternative dual-grid approach hin 2- and d3d 3-d. Explore alternative methods for incorporating an arbitrary boundary geometry into the Material Point Method. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 47
Thank you! All workshop participants PI P.I. s Peter Mackenzie-Helnwein, i Pedro Arduino, and Greg Miller. The National Science Foundation grant CMMI-0900318 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 48
QUESTIONS???? 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 49
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Approach Dual-Grid methodology The Enhanced Velocity Field Approach The total velocity and acceleration field exists as a superposition from both grids.... With the conditions for 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 52
Approach Ω A Ω α Γ The Enhanced Velocity Field Approach Enforce essential condition along Γ Leads to a constraint of the form 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 53
Approach The Enhanced Velocity Field Approach Algorithmic implementation: 1. Obtain the nodal acceleration at time t n for those nodes in the boundary grid as well as the standard grid by solving the system. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 54
Approach The Enhanced Velocity Field Approach Algorithmic implementation: 2. Obtain the nodal velocity at time t n for those nodes in the boundary grid as well as the standard grid by solving the system. 3. Update nodal values for both grids. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 55
Approach The Enhanced Velocity Field Approach Algorithmic implementation: 4. Update all particles using the nodes on the standard grid. 5. For those particles with p, perform an additional update using those nodes in the boundary grid. 8/9/2010 Carter Mast - University of Washington - 6th MPM Workshop 56
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