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[[Category: Users:Structural Optimization]] == General Description == === Short Info === In the field of structural optimization it is often necessary to apply a constraining of surface curvature in order to maintain manufacturing constraints. To this purpose, ''Carat++'' provides an estimation tool to approximate the mean curvature at a surface node. === Estimation of nodal curvature === The nodal curvature is estimated using the surface normal vector and access vectors to the surrounding nodes. '''The estimation is based on a local spherical approximation of the geometry (see sketch below).''' So the radius of curvature r can be computed within the isosceles triangle. [[ File:curv_ks_sketch.png |500px | center |Sketch of curvature estimation in 1D ]] Having computed this approximated radius of curvature for each surrounding node, the mean curvature of the surface is estimated by a weighted sum over all surrounding nodes ''i'', whereat v''i'' is the access vector to the ith node. [[ File:curv_ks_formula.png |250px | center | ]] === Kreisselmeier-Steinhauser weighting === Instead of constraining each single nodal curvature, ''Carat++'' provides a technique to control the curvature within bigger element patches. To this purpose a Kreisselmeier-Steinhauser weighting is used. [[ File:curv_ks_formula2.png |350px | center | KS formulation for one patch ]] Via the weighting parameter ρ the user can control to which extend exceedings of the maximum allowable curvature are weighted. == Input Parameters == {| border="1" cellpadding="3" cellspacing="0" |colspan="3" style="background:#efefef;"| Block headline |- !Parameter !Values, Default(*) !Description |- !OPT-RESPONSE_FCT |''int'' : CURVATURE_PATCH_KS |Function ID and response function type. |- {{Users:Structural Optimization/Response Functions/Common Parameters}} |- |colspan="3" style="background:#efefef;"| Specific parameters |- !NODES |ND-SET ''int'' |Node set that includes all nodes this response function yields onto. |- !PATCH_SIZE |''int'' |Set of ''NODES'' is sub-divided in patches with ''PATCH_SIZE'' nodes in each patch. A single response function is created for each of these patches. |- !RHO |''real'' |KS-weighting factor. |- {{Users:Structural Optimization/Response Functions/Common Parameters for Constraints}} |- |} === Example of a Complete Input Block === <pre> OPT-RESPONSE_FCT 1 : CURVATURE_PATCH_KS WEIGHT=1.0 ETA=1e-06 GRAD=ADJOINT SA= SEMI_ANALYTIC FDA=FOREWARD DESVAR=OPT-VAR 1 ! -- specific parameters NODES=ND-SET 1 PATCH_SIZE = 5 RHO = 15 ! -- constraint parameters ABS_LIMIT = 0.25 ! KAPPA_MAX REL_TOLERANCE = 0.1 CONSTRAINT_TYPE = INEQUALITY_LT LAMBDA_ABS_MAX = 40 </pre> == A complete test example == === Model description === The considered test example is the optimization of a pinched cylinder applying a curvature limitation. [[File:RespFunc_curvKS_system.jpg | center | 300px | pinched cylinder]] The curvature is controlled via a ''CURVATURE_PATCH_KS''-function using a patch-size of 10 and a rho-parameter of 15. The kappa_max value is set to 7.0. === Input File === Here the full input file can be downloaded. === Documented Results === Based onto the user input, ''Carat++'' automatically generated 22 patches and generated a constraint function for each of them. After 32 optimization steps all constraints are fulfilled and the maximum nodal curvature appearing is 8.4. <pre> 10:18:41 -------------------------------------------------------------------------------------- 10:18:41 ID | type | active | lambda | value | tol | psi 10:18:41 20001 inequality yes 0.170140 -0.006070 0.050000 -0.006070 10:18:41 20002 inequality no 0.000000 -0.338485 0.050000 0.000000 10:18:41 20003 inequality no 0.000000 -0.291237 0.050000 0.000000 10:18:41 20004 inequality yes 5.573192 -0.028454 0.050000 -0.028454 10:18:41 20005 inequality no 0.000000 -0.399155 0.050000 0.000000 10:18:41 20006 inequality no 0.000000 -0.399018 0.050000 0.000000 10:18:41 20007 inequality no 0.000000 -0.701880 0.050000 0.000000 10:18:41 20008 inequality no 0.000000 -0.701884 0.050000 0.000000 10:18:41 20009 inequality no 0.000000 -0.708979 0.050000 0.000000 10:18:41 20010 inequality no 0.000000 -0.708979 0.050000 0.000000 10:18:41 20011 inequality no 0.000000 -0.687952 0.050000 0.000000 10:18:41 20012 inequality no 0.000000 -0.687946 0.050000 0.000000 10:18:41 20013 inequality no 0.000000 -0.695694 0.050000 0.000000 10:18:41 20014 inequality no 0.000000 -0.695704 0.050000 0.000000 10:18:41 20015 inequality no 0.000000 -0.257635 0.050000 0.000000 10:18:41 20016 inequality no 0.000000 -0.258434 0.050000 0.000000 10:18:41 20017 inequality no 0.000000 -0.253883 0.050000 0.000000 10:18:41 20018 inequality no 0.000000 -0.253481 0.050000 0.000000 10:18:41 20019 inequality yes 0.266234 0.010527 0.050000 0.010527 10:18:41 20020 inequality yes 0.000000 -0.039598 0.050000 -0.019176 10:18:41 20021 inequality yes 0.252340 -0.012201 0.050000 -0.012201 10:18:41 20022 inequality yes 0.000000 -0.033469 0.050000 -0.016060 10:18:41 ----------------------------------------------------------- 10:18:41 Optimization step 32 finished! 10:18:41 ----------------------------------------------------------- </pre> [[File:RespFunc_curvKS_res.png | center | 300px | nodal curvature]]

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