UNIVERSITY Stress constrained optimization using XFEM and Level

UNIVERSITY Stress constrained optimization using X-FEM and Level Set Description of Liège L. Van Miegroet, T. Jacobs E. Lemaire and P. Duysinx Automotive Engineering / Multidisciplinary Optimization Aerospace and Mechanics Department University of Liège

n Introduction UNIVERSITY n e. Xtended Finite Element Method (XFEM) n Level Set Method of Liège Stress constrained optimization using X-FEM and Level Set Description Outline n Sensitivity Analysis n Applications n Conclusion INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY n Shape optimization : n Variables : geometric parameters n Limitation on shape modification n Mesh perturbation – Velocity field n CAD model of structure n Various formulations available of Liège Stress constrained optimization using X-FEM and Level Set Description Introduction – Optimization of mechanical structures n Topology optimization : n Variables : distribution of material element densities n Strong performance – deep change allowed n Fixed Mesh n « Image » of the structure n Limitation of objective function and constraints INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY n X-FEM and Level Set optimization XFEM + Level Set description = Intermediate approach between shape and topology optimisation n e. Xtended Finite Element Method Alternative to remeshing n work on fixed mesh n no velocity field and no mesh perturbation required n Level Set Method Alternative description to parametric CAD n Smooth curve description n Topology can be altered as entities can be merged or separated generalized shape n Problem formulation: n Global and local constraints n Small number of design variables of Liège Stress constrained optimization using X-FEM and Level Set Description Introduction – n INTRODUCTION Introduction of new holes requires a topological derivatives XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description e. Xtended Finite Element Method n Early motivation : n Study of propagating crack in mechanical structures avoid the remeshing procedure n Principle : n Allow the model to handle discontinuities that are non conforming with the mesh n Representing holes or material – void interfaces n Use special shape functions V(x)Ni(x) (discontinuous) INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n Principle (Sethian & Osher, 1999) UNIVERSITY of n n n Introduce a higher dimension Implicit representation Interface = the zero level of a function : n Possible practical implementation: Liège Stress constrained optimization using X-FEM and Level Set Description The Level Set Description n Approximated on a fixed mesh by the signed distance function to curve G: n Advantages: n 2 D / 3 D n Combination of entities: e. g. min / max INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n 1. Build Level Set n 2. Detect type of element n 3. Cut the mesh n 4. Working mesh for integration UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description The Level Set and the X-FEM INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n Geometry description and material layout: UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description Optimization Problem Formulation n Using Basic Level Sets description : ellipses, rectangles, NURBS, … n Design Problem: n Find the best shape to minimize a given objective functions while satisfying design constraints n Design variables: n Parameters of Level Sets n Objective and constraints: n n Mechanical responses: compliance, displacement, stress, … Geometrical characteristics: volume, distance, … n Problem formulation similar to shape optimization but simplified thanks to XFEM and Level Set! INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

principles n Classical approach for sensitivity analysis in industrial codes: UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description Sensitivity analysis - semi analytical approach based on K derivative n Discretized equilibrium: n Derivatives of displacement: n Semi analytical approach: INTRODUCTION n Stiffness matrix derivative: n Compliance derivative: n Stress derivative: XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

technical difficulty n Introduction of new dofs K dimension changes not possible UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description Sensitivity analysis – n INTRODUCTION Ignore the new elements that become solid or partly solid n Small errors, but minor contributions n Practically, no problem observed n Efficiency and simplicity n Validated on benchmarks XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY validation n Validation of semi-analytic sensitivity: n Elliptical hole n Parameters: major axis a and Orientation angle q w. r. to horizontal axis n of Perturbation: d=10 -4 Liège Stress constrained optimization using X-FEM and Level Set Description Sensitivity analysis – INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

2 D plate with a hole n Plate with generalized super elliptical hole : UNIVERSITY n Parameters : n Objective: min Compliance. Constraint: upper bound on the Volume. n of Liège Stress constrained optimization using X-FEM and Level Set Description Applications - INTRODUCTION n n Bi-axial Load: Solution: perfect circle: XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

2 D fillet in tension n Shape of the fillet : generalized super ellipse UNIVERSITY of Liège Stress constrained optimization using X-FEM and Level Set Description Applications – INTRODUCTION n Parameters : n Objective: min (max Stress) No Constraint Uni-axial Load: Solution: stress reduction of 30% n n n XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

2 D plate with elliptical hole n Plate with elliptical hole : UNIVERSITY of n n n Parameters : Objective: min Compliance. Constraint: s. VM <= 0. 8 s. VM 0 Bi-axial Load: Solution: perfect circle. s. VM 0 : 2. 16 s. VM final: 1. 68 Liège Stress constrained optimization using X-FEM and Level Set Description Applications – INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n XFEM and Level Set gives rise to a generalized shape UNIVERSITY optimisation technique n Intermediate to shape and topology optimisation of n n INTRODUCTION Work on a fixed mesh Topology can be modified: n Liège Stress constrained optimization using X-FEM and Level Set Description Conclusion n n n Holes can merge and disappear New holes cannot be introduced without topological derivatives Smooth curves description Void-solid description Small number of design variables Global or local response constraints No velocity field and mesh perturbation problems XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n Contribution of this work UNIVERSITY n n n New perspectives of XFEM and Level Set Investigation of semi-analytical approach for sensitivity analysis Implementation in a general C++ multiphysics FE code of Liège Stress constrained optimization using X-FEM and Level Set Description Conclusion n Perspectives: INTRODUCTION n n n 3 D problems Dynamic problems Multiphysic simulation problems with free interfaces XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY n Representing holes or material – void interfaces n Remove empty elements n Keep partially filled elements (cut element) n Use X-FEM numerical integration no contribution of void part of Liège Stress constrained optimization using X-FEM and Level Set Description e. Xtended Finite Element Method INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

n Other Strategies to freeze the number of dof : UNIVERSITY of n analytical derivatives of stiffness matrix not general! n boundary layer in which all elements are retained n rigid modes, larger size of the problem Liège Stress constrained optimization using X-FEM and Level Set Description Sensitivity analysis INTRODUCTION n boundary layer with softening material (SIMP law) n n lost of void / solid approximation Introduce the contribution of existing nodes XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION

UNIVERSITY Perturbation of Liège Stress constrained optimization using X-FEM and Level Set Description Sensitivity analysis n No contribution of node 4 Perturbation INTRODUCTION XFEM LEVEL SET SENSITIVITY APPLICATIONS CONCLUSION