Performance Evaluation of EdgeBased Smoothed Finite Element Method

Performance Evaluation of Edge-Based Smoothed Finite Element Method for 4 -node Tetrahedral Meshes on Electrodeposition Simulation Kai KITAMURA(1), Yuki ONISHI(1), Takeshi KASHIYAMA(2), Kenji AMAYA(1) Tokyo Institute of Technology (Japan) (2) SUZUKI MOTOR CORPORATION (Japan) ICCM 2019 P. 1

What is electrodeposition (ED) ? + - Paint ＋ An o d e ＋ Cathode http: //n-link. nissan. co. jp/NOM/PLANT/ n Most widely-used basecoat methods for car bodies. n Making coated film by applying direct electric current in a paint pool. n Relatively good at making uniform film thickness but not satisfactory uniform in actual production lines. n ED simulator is necessary for the optimization of carbody design and coating conditions in actual lines. ICCM 2019 P. 2

Photos of ED process line 2. water rinse process 1. dipping and deposition process We focus on this process. 3. baking process ICCM 2019 P. 3

Mechanism of Electrodeposition Voltage is applied Deposition - Electrophoresis + + Paint Ions Carbody （Cathode) - + Aggregated into paint particles - When the particle gets big enough Coated Film n. Positively charged paint ions are attracted to the cathode. n. Paint ions lose their electrical charge and are aggregated into paint particles. n. Some of the paint particles diffuse and dissolve. ICCM 2019 P. 4

What is ED Simulation ? ED simulation provides film thickness, surface potential, surface current density and so on. Film Thickness You can see that the paint film starts to be deposited from the outside surface. Surface Potential Surface Current Density ICCM 2019 P. 5

How to Develop an ED Simulator 1. Experiments at lab in various coating conditions. One-Plate Test 2. Identification of ED boundary model and its parameters. Plate 3. Implementation to a FE code. Paint On the analogy of solid mechanics, Step 1: material tests DELIGHT web page, with MTS, https: //delightcfd. com Step 2: identification of elastoplastic model. ED simulation for actual lines ICCM 2019 P. 6

Issues in Meshing (1) ✗ It is difficult to discretize complex shapes such as car bodies with hexahedral meshes. ✗ Only surface mesh is shown. → We have to use tetrahedral meshes in ED simulation. 　　However… Accuracy of the standard FEM-T 4 is insufficient in complex shapes. ICCM 2019 P. 7

Issues in Meshing (2) ✗ 10 -node tetrahedral (T 10) mesh without kink generally requires more large number of nodes than T 4 mesh. 　　　 T 4 　　 T 10 without kink Carbody hole For the same shape representation, T 10 mesh without kink leads to massive increase in DOF. ICCM 2019 P. 8

Issues in Meshing (2 Cont. ) ✗ 10 -node tetrahedral (T 10) mesh with kink causes severe accuracy loss. 　　　 T 4 　　 T 10 with kink Carbody Kinked T 10 hole T 10 mesh with kink does not increase DOF but induces severe accuracy loss. ICCM 2019 P. 9

Motivation n Hexahedral elements: 　✗ It is difficult to discretize complex shapes. n T 10 elements without kink: 　✗ It leads to massive increase in DOF. n T 10 elements with kink: 　✗ It causes severe accuracy loss. → We want to realize high accuracy analysis with T 4 mesh. ES-FEM-T 4 could be a solution to these issues. ICCM 2019 P. 10

Objective Development of ED simulator using ES-FEM-T 4 and its performance evaluation by comparing with FEM-T 4. Table of body contents: 1. Outline of ED Simulation 2. Formulation of ES-FEM for ED Simulation 3. Analysis Results ICCM 2019 P. 11

Outline of ED Simulation ICCM 2019 P. 12

n Fundamental Equations + - Paint An o d e ＋ Cathode ＋ Solving the Laplace equation for potential, the current density distribution on a carbody is determined and then the film thickness distribution is time-evolutionally calculated. ICCM 2019 P. 13

Two Complexities of ED Phenomena n ICCM 2019 P. 14

Procedure to Identify Film Resistance Model 1. One-plate tests Plot 3. Approximated surface Data fitting ICCM 2019 P. 15

Procedure to Identify Film Growth Model 1. One-plate tests Plot Diffusive current density represents the amount of electricity consumed for the electrolysis of water, not used for the deposition of the coating film. 3. Approximated surface Data fitting ICCM 2019 P. 16

Formulation of ES-FEM for ED Simulation ICCM 2019 P. 17

Outline of ES-FEM What is ES-FEM-T 4? n n A kind of strain smoothing method. Using element edges as Gauss points. Robust against element skew. Super-linear mesh convergence rate with T 4 mesh. Standard FEM ES-FEM assembles each element’s value. ES-FEM assembles each edge’s value. Integration domain Element Node ICCM 2019 P. 18 Integration domain is different !!

Outline of ES-FEM Short edges n When meshing complex shape, the generation of short edges is inevitable. n Short edges lead to accuracy loss in standard FEM-T 4. Conventional FEM 　 ES-FEM can suppress the accuracy loss by smoothing. ICCM 2019 P. 19

Analysis Results ICCM 2019 P. 20

Outline 4 -Plate BOX Simulation Film Thickness Side Sill 4 -P BOX (http: //plaza. rakuten. co. jp/heroc) n Imitating a bag-like structure such as side sill in a carbody. n Accuracy on the innermost surface (leftmost plate surface) is the most important; i. e. , “maximize the minimum”. n Film thickness is calculated with 4 different mesh seed sizes and compared between FEM-T 4 and ES-FEM-T 4. ICCM 2019 P. 21

4 -Plate BOX Simulation Overview of Meshes Only the surface meshes are shown. 3. 2 mm Mesh Seed Size (31 k T 4 elem. ) 1. 6 mm Mesh Seed Size (65 k T 4 elem. ) 0. 8 mm Mesh Seed Size (169 k T 4 elem. ) 0. 4 mm Mesh Seed Size (716 k T 4 elem. ) ICCM 2019 P. 22

4 -Plate BOX Simulation Film Thickness of A-Plate (outermost surface) Mesh seed size FEM results (dashed lines) have tiny errors tiny due to mesh coarseness. Meanwhile, ES-FEM (solid lines) results have no such errors. ICCM 2019 P. 23

4 -Plate BOX Simulation Film Thickness of C-Plate (surface in the 1 st bag) Mesh seed size FEM results (dashed lines) have small errors small due to mesh coarseness. Meanwhile, ES-FEM (solid lines) results have no such errors. ICCM 2019 P. 24

4 -Plate BOX Simulation Film Thickness of E-Plate (surface in the 2 nd bag) Mesh seed size FEM results (dashed lines) have medium errors medium due to mesh coarseness. Meanwhile, ES-FEM (solid lines) results have no such errors. ICCM 2019 P. 25

4 -Plate BOX Simulation Film Thickness of G-Plate (innermost surface) Mesh seed size FEM results (dashed lines) have large errors large due to mesh coarseness. Meanwhile, ES-FEM (solid lines) results have no such errors. ICCM 2019 P. 26

4 -Plate BOX Simulation Error of Final Film Thickness on G-Plate The result of ES-FEM with the minimum mesh seed size (0. 4 mm) is used as the reference. ES-FEM-T 4 has far better mesh convergence rate than FEM-T 4 !! ICCM 2019 P. 27

Outline Carbody Simulation 270 V 0 30 180 (s) n A half carbody is fixed in a box pool. n The side wall is treated as an anode surface. n Compare the time-developed film thickness between FEM-T 4 and ES-FEM-T 4 with a same mesh. ICCM 2019 P. 28

Carbody Simulation Overview of Surface Mesh Provided by SUZUKI MOTOR Co. n 13 M T 4 elements (3 M nodes & 18 M edges) in total in the pool. ICCM 2019 P. 29

Outer View Carbody Simulation FEM-T 4 ES-FEM-T 4 There is no mush difference on the outer surfaces. ICCM 2019 P. 30

Clipped View on Side Sill Carbody Simulation FEM-T 4 As the ES-FEM-T 4 4 -P BOX case, ES-FEM-T 4 presents thinner dist. on the side sill. Big difference appears on the inner surfaces. ICCM 2019 P. 31

Comparison of Computational Costs Calculation Time on a PC with Intel i 9 -9960 X using 10 cores ES-FEM-T 4 0. 02 h 0. 05 h 0. 45 h 4 -P BOX with 0. 4 mm mesh Carbody 9. 5 h 67 h 9. 0 h 125 h S Ac am cu e ra cy 4 -P BOX with 3. 2 mm mesh 4 -P BOX with 1. 6 mm mesh 4 -P BOX with 0. 8 mm mesh FEM-T 4 0. 02 h 0. 04 h 0. 45 h There is no big difference in calculation time although the accuracy of ES-FEM-T 4 is much better. ICCM 2019 P. 32

Summary ICCM 2019 P. 33

Summary Conclusion n ES-FEM-T 4 was applied to ED simulations. n High accuracy of ES-FEM-T 4 because of its superlinear mesh convergence rate was confirmed in comparison to the poor accuracy of FEM-T 4. Future Works n Validation of the ED models on the actual manufacturing lines. n Calculation speed-up with distributed memory parallelization. Thank you for your kind attention. ICCM 2019 P. 34

Appendix ICCM 2019 P. 35

ED Boundary Models ICCM 2019 P. 36

Comparison of Computational Costs This is because … n Most of the calculation time is consumed by the iterative matrix solver (MINRES with Jacobi preconditioner). n The matrix band width of ES-FEM-T 4 is 3 times wider than that of FEM-T 4; i. e. , ES-FEM-T 4 requires 3 times larger memory size. n However, the iteration count of MINRES in ES-FEMT 4 is about 1/3 or 2/3 of that in FEM-T 4 thanks to the well-posedness of the matrix. ICCM 2019 P. 37

Carbody Simulation Surface Current Density FEM-T 4 　　ES-FEM-T 4 ES-FEM suppresses the spike error of surface current density appearing in FEM. ICCM 2019 P. 38

Carbody Simulation Surface Current Density FEM-T 4 　　ES-FEM-T 4 ES-FEM suppresses the spike error of surface current density appearing in FEM. ICCM 2019 P. 39

Carbody Simulation Surface Current Density FEM-T 4 　　ES-FEM-T 4 ES-FEM suppresses the spike error of surface current density appearing in FEM. ICCM 2019 P. 40