Robustness improvement of polyhedral mesh method for airbag deployment simulations. TU Delft
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1 Robustness improvement of polyhedral mesh method for airbag deployment simulations. TU Delft Santiago Alagon Carrillo, Numerical Analysis Daily Supervisor: Prof. dr. ir. C. Vuik, Company Supervisors: Dr. M. Lewis
2 Outline Industry Background and Problem Description Mathematical tools and Results Solution Methods. Implementation and results Contributions and Conclusion Questions
3 Industry Background and Problem Description TASS develops software to analyze occupant safety systems. MADYMO solver platform comprises Multi-Body, Finite-Element, Computational Fluid Dynamics and Contact Models. MADYMO is constantly being updated. CFD solver: Gasflow Gasflow2 robustness issues
4 Industry Background and Problem Description Airbag flexible membrane. Modeled by equations of elasticity, solved numerically using FEM Flow Euler Equations, solved numerically using FVM. Use of Cartesian Mesh Method.
5 Industry Background and Problem Description Airbag triangulation intersects in arbitrary way the Cartesian Mesh. Cartesian cells classifies as active, inactive and intersected cells. Gasflow staircase implementation. Gasflow2 cutcell implementation.
6 Industry Background and Problem Description
7 Geometrical Algorithm. CFD solver requires the geometrical and topological information of the Cartesian Cells. Search Algorithm obtains exact geometry. Coloring Algorithm Classifies the cells into active cells, inactive cells and cut-cells; and determines the active and inactive regions of the cut-cells.
8 Coloring Algorithm and Problem Description Project Goal: design the Coloring Algorithm for a Cartesian cut-cell method suitable for time varying flow domains defined by a Finite Element triangulation. Focus of the project placed on the determination of the active and inactive regions of the cut-cells.
9 Search Algorithm. Geometrical constructors too expensive! Signed volume of a simplex va 0 va 1 1 v b 0 vb 1 V (Tv0 v1...vd ) = 3! vc0 vc1 vd 0 v d 1 va 2 vb 2 vc 2 vd Figuur: Signed Volume Property and Topological Tests (1)
10 Search Algorithm. Global Search
11 Search Algorithm. Euler Search and Exact Geometry Search
12 Mathematical tools and Results What s the big deal? Airbag sliced by plane results in polygon. Cell Face is a polygon. The intersection of the airbag polygon and cell face polygon results in the active facepolygon.
13 Abstract Geometrical Modeling. Representation of polygons using Simplicial Chains. A simplex is a generalization of the notion triangle to any dimension. A d-dimensional simplicial chain is an expression of the form λ = m α i S i where the coefficients α i are integers and the S i are simple d-dimensional simplices. i=1
14 Abstract Geometrical Modeling. Theorem Let P be a bidimensional polyhedral solid determined by n edges e 1, e 2,...e n oriented in counterclockwise order. Let S i be the simplex determined by the origin O and the edge e i,beingv i1 and V i2 their vertices and α i its coefficients. Then P = P λ where λ = n α i S i and α i = sign(area sign(t i )) i=1
15 Abstract Geometrical Modeling. Theorem Let P 1 and P 2 be polyhedral solids in R d,andχ 1 and χ 2 their associated chains respectively χ 1 = n m α i S i, χ 2 = β i U i i=1 j=1 Then the associated normal chain to the intersection solid P χ = P 1 P 2 is χ = n m (α i β j ) Sim(S i U j ) i=1 j=1
16 Abstract Geometrical Modeling. Result The polygonal regions inside each cut cell face, P C, belonging to the flow regions are those for which the counterclockwise description of their boundaries from, form the point of view of an interior point to the region is completely consistent with the general counterclockwise descriptions of the polygons P A and P C.
17 Two dimensional equivalence All observations are performed from the point of view of an observer inside the cut cell. Figuur: View point for cut cell
18 Problematic cases If we describe all polygons in CCW fashion, are we done then? Not quite! Figuur: Cases where no niccw can be determined.
19 Non Local This method would require non-local information Figuur: Cell face plane regions
20 Problematic cases The problem of non-local information can be solved using the outward normal of the FE-triangulation Still there are problems
21 Test case classification. Test case 1, At most one zero and only s-face segments Figuur: Test case 1 Test case 2 and 3, At most two zeros. Figuur: Test case 2 and 3.
22 Test case classification. Test case 4, Three zeros, coplanar case Figuur: Test case 4 Test case 5, only one zero, cone in one side of plane. Figuur: Test case 5.
23 Test case classification. Test case 6, 7, 8 and 9, only one zero, no face segments Figuur: Test case 6,7,8,9.
24 Solution methods Extended Consistency Method, for test cases 1, 3 and 4 Figuur: Test case 1. For each cut-cell face, traverse each of the face polygons in counter-clock wise direction. For each of the polygons, if the counter-clock wise direction is not consistent with niccw direction of at least one of the face segments then the face polygon is determined to be outside.
25 Solution methods Extended Consistency Method, for test cases 1, 3 and 4 Can be used to solve test cases 3 and 4 through the projection of the edge-outward pointing normal ˆn ˆn = ˆn k +ˆn l ˆn k +ˆn l (2) Figuur: Traversing direction consistency..
26 Solution methods Extended Consistency Method, for test cases 1, 3 and 4 Application to simple polygons and contained polygons Figuur: Traversing direction consistency..
27 Solution methods Local half-space determination method 1, treatment of Test Case 3 Project the edge-outward pointing normal ˆn onto the face outward pointing normal. Verify the sign of the projection. Non problematic case Problematic case Figuur: Projection of edge-outward normal.
28 Solution methods Local half-space determination method 2, treatment of Test Case 5 Project the outward pointing normal of the FE-element closes to the cell face plane onto the face outward pointing normal. Verify the sign of the projection. Figuur: Projection of edge-outward normal.
29 Solution methods Local half-space determination method 3, treatment of Test Cases 6, 7, 8 and 9 Compute all zeros of FE-elements and cell face plane Build edges cs i between the computed zeros and the polypoint Find closest cs i edge to cell edge ce i Project outward pointing normal of cs i onto outward pointing normal of ce i and ce i+1 Use the sign of these projections to determine activity of cell face Figuur: Projection of edge-outward normal
30 Test case classification. Test case 6, 7, 8 and 9, only one zero, no face segments can be treated the same way Figuur: Test case 6,7,8,9.
31 Implementation Implementation requirements. Object oriented programming for the implementation. Each geometric object is defined in terms of a programing object with properties. Implementation using a single test cell for each solution method. Each cell contains all the necessary geometric information obtained from the Exact Geometry search.
32 Implementation Implementation: Extended Consistency Method. Figuur: Test case for Extended Consistency Method. Figuur: Results for Extended Consistency Method
33 Implementation Implementation: Half-space Determination Method 2. Figuur: Test case for Half-space Determination Method 2. Figuur: Results for Half-space Determination Method 2.
34 Implementation Implementation: Half-space Determination Method 3. Figuur: Test case for Half-space Determination Method 3. Figuur: Results for Half-space Determination Method 3.
35 Results Test cut cell representative of all possible geometries according to classification. Implementation of the different solution successfully determines the active and inactive face polygons Coloring algorithm capable of determining the active and inactive regions of all possible geometries of cut cells.
36 Conclusion A coloring algorithm was developed which successfully determines the active and inactive regions of the cut cells. This Algorithms has the following characteristics: Is based on a classification of all possible geometries of the cut cells from which four types of cut cells are obtained, each treated with a different method. The determination of the activity of the face polygons uses only local information available to each cell. Suitable for parallel implementation. Determination of the activity of each face polygon based only on topological tests.
37 QUESTIONS
Robustness improvement of polyhedral mesh method for airbag deployment simulations. TU Delft
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