L5.1w w w .3d s .c o m | © D a s s a u l t S y s t èm e sLesson content:Problem StatementTopology Optimization – ResultsTopology Optimization – Results Examination Topology Optimization – Analysis ConclusionsLesson 5: Nonlinear Geometric Effects in Topology Optimization30 minutesL5.2w w w .3d s .c o m | © D a s s a u l t S y s t èm e sProblem StatementConsider a beam structure, clamped at both ends, subjected to a prescribed displacement in its center region.Topology optimization task:Minimize the strain energy while using only 10% of the original mass.Evaluated solver and material combinations:Linear geometry and linear material Linear geometry and nonlinear material Nonlinear geometry and linear material Nonlinear geometry and nonlinear materialprescribed displacementPrescribe displacementc l a m p ede n ds y m m e t r yMechanical model Finite element model, exploiting symmetryw w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology optimization - Evolving geometriesLinear geometry + Linear material Nonlinear geometry + Linear materialLinear geometry + Nonlinear materialNonlinear geometry + Nonlinear materialL5.4w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results (2/5)Final configurationWhat happened here?What happened here?Linear geometry + Linear materialNonlinear geometry + Linear materialLinear geometry + Nonlinear material Nonlinear geometry + Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sDisplacements - Final configurationLinear geometry + Linear materialNonlinear geometry + Linear materialLinear geometry + Nonlinear material Nonlinear geometry + Nonlinear materialL5.6w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results (4/5)Von Mises Stress - Final configurationLinear geometry + Linear materialNonlinear geometry + Linear materialLinear geometry + Nonlinear material Nonlinear geometry + Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sFollowing the “classic” first steps in examining FE simulation results, looking at displacements and von Mises stresses do not indicate that something could be wrong here!A closer examination of the results reveals important differences!L5.8w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results Examination (1/7)Optimization Step 0 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sOptimization Step 4 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialL5.10w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results Examination (3/7)Optimization Step 10 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sOptimization Step 12 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialL5.12w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results Examination (5/7)Optimization Step 13 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sOptimization Step 14 - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialL5.14w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Results Examination (7/7)Optimization Step 15 (final) - Principal stressesLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Nonlinear material Nonlinear geometry Nonlinear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sWhat happened here?For this model, the essential changes occur between optimization steps 12 and 13 (shown on next slide)As material is removed from the domain, the loading in the center region changes from shear dominated (up to step 12) to bending dominated (step 12 and onwards).The geometrically linear model does not “sense” this important change, causing it to evolve into a compression member model.The geometrically nonlinear model correctly picks up the change to bending causing it to evolve into a tension member model.Material nonlinearity did not play a role in this model.L5.16w w w .3d s .c o m | © D a s s a u l t S y s t èm e sTopology Optimization – Analysis (2/3)Optimization Step 12 - Principal stressesOptimization Step 13 - Principal stressesLinear geometry Linear material Nonlinear geometry Linear materialLinear geometry Linear materialNonlinear geometry Linear materialw w w .3d s .c o m | © D a s s a u l t S y s t èm e sPlastic strain- Final configurationLinear geometry Linear materialNonlinear geometry Linear materialLinear geometry Linear material Nonlinear geometry Linear materialL5.18w w w .3d s .c o m | © D a s s a u l t S y s t èm e sConclusionsDo nonlinear geometric effects really matter? YESComparing the final optimization result:The compression member solution is mechanically unstable !An increase above the design load will cause the structure to snap through and fail!Linear geometry Linear material Nonlinear geometry Linear material。