SSLL117 Validation of modelings second gradient

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1 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 1/13 Responsable : FRNANDS Roméo Clé : V Révision : SSLL117 Validation of modelings second gradie Summary: This test makes it possible to validate modelings second gradient [R ] while being based on analytical solutions. Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

2 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 2/13 Responsable : FRNANDS Roméo Clé : V Révision : 1 Problem of reference 1.1 Geometry Figure 1.1-a Coordinates of the pois X Y A B C D The geometry of the structure is a square length of with dimensions 1m. The test is carried out on only one isoparametric finite eleme of quadratic form, named group of meshs ROCH. The various sides of this square, useful for this modeling, are groups of meshs, DA and BC. The group of meshs ROCH, coains the groups of meshs in addition SROCH and MROCH ; who correspond respectively to the nodes tops and mediums of this same group. 1.2 Properties of materials In the coext of the mediums of the second gradie, it is necessary to define properties materials for the parts attached to the first and second gradies of the field of displaceme. The properties of material are elastic for these two parts: Young modulus: =1000 Pa ; Poisson's ratio: =0 ; Microscopic module of rigidity: a 1 =10 Pa.m 2. Parameter of penalization: r=18 Parameters material of the mediums of the second gradie: a 2 =a 3 =a 4 =a 5 =0Pa.m Boundary conditions and loadings Dy=0 everywhere (modeling is brought back in the case 1D ) Dx=0 on the segme BC Dx=0.1m on the segme AD Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

3 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 3/13 Responsable : FRNANDS Roméo Clé : V Révision : 2 Reference solution 2.1 Method of calculating The problem to be solved is: u with u,, We have: x x u u x x r u x u x kinematically acceptable. u { = x = F x =0 Let us note functions of form of the second order and N functions of first order form on the linear elemes, Let us recall by the same occasion the definitions of SG2 and SG3: SG2 : segme with 2 nodes many nodes: 2 many nodes tops: 2 SG3 : segme with 3 nodes many nodes: 3 many nodes tops: 2 Figure 2.1-a Figure 2.1-b N 1 = 1 x and N 2 2 = 1 x 2 form with the first node. on the eleme of reference x [ 1, 1 ] with N 1 the function of The functions of form of the segme to the 3 nodes are then: 1 = x 1 x 2 2 = x 1 x and 2 3 =1 x 2 Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

4 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 4/13 Responsable : FRNANDS Roméo Clé : V Révision : While posing u = =0 we show that: 1 1 u x dx=0 however u=.u.u.u with u 2 =0 ( C.L. ) from where u x = x 1 2 u 1 2 x u 3 and v=n 1. 1 N 2. 2 with 2 =0 ( C.L. ) 1 x We thus have = 2 1 however, it should be noted after iegration that: 1 = u 1 (1) u,, kinematically acceptable. While proceeding in the same way and while posing: = =0 we find that: u 3 = 2 r 4 r.u 1 u 1 4 (2) from where the general formula S 1 =3.a 1. x = 3 2. a1. 1 and after simplification S 1 = 3 2.a1.u Sizes and results of reference The fundameal and very general formula for the solid is S 1 = 3 2.a1.u Uncertaiies on the solution Analytical solution. Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

5 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 5/13 Responsable : FRNANDS Roméo Clé : V Révision : 3 Modeling A 3.1 Characteristics of modeling The characteristics are ideical to the reference solution. Modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_DIL with the choice of ierpolation P2-P1-P Characteristics of the grid The group of meshs ROCH_RG is obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 8 Number of SG3 4 Number of QUAD8 2 Number of group of meshs Sizes tested and results Value tested Mome Node Reference Criterion Aster Tolerance Displaceme GONF 1.0 N1 NON_RGRSSION RLATIV ,010 Displaceme DX 1.0 N5 NON_RGRSSION RLATIV ,010 Force of reaction SIG1 1.0 NON_RGRSSION RLATIV 1.5 0,010 Force of reaction DPV 1.0 NON_RGRSSION ABSOLUT Remarks With an aim of validating the option of resolution RIGI_MCA_LAS of STAT_NON_LIN, a calculation ideical to the first (option MATRIC=TANGNT ) is realized by imposing this option of resolution. The results are rigorously ideical to those obtained with the first option of calculation which are preseed in the table of results above. Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

6 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 6/13 Responsable : FRNANDS Roméo Clé : V Révision : 4 Modeling B 4.1 Characteristics of modeling In this case, the parameters materials resulting from Commandes tests of nonregression generated with code source are ideical in two modelings A and B, only differs the quadratic grid. The objective of this modeling is to have a validation on a structure with a grid, the analytical solution being difficult to obtain and consequely the solution in not-regression, is privileged here. Modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_DIL with the choice of ierpolation P2-P1-P Characteristics of the grid The group of meshs ROCH_RG is obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 341 Number of SG3 40 Number of QUAD8 200 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme GONF 1.0 N1 NON-RGRSSION RLATIV Displaceme DX 1.0 N5 NON-RGRSSION RLATIV Force of reaction DPV 1.0 NON-RGRSSION ABSOLUT Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

7 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 7/13 Responsable : FRNANDS Roméo Clé : V Révision : 5 Modeling C 5.1 Characteristics of modeling This modeling is ideical to modeling A. modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_DIL with the choice of ierpolation P2-P1-P Characteristics of the grid The group of meshs ROCH_RG St obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 9 Number of SG3 4 Number of TRIA6 4 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme GONF 1.0 N1 ANALYTICAL RLATIV ,010 Displaceme DX 1.0 N5 ANALYTICAL RLATIV ,010 Force of reaction SIG1 1.0 ANALYTICAL RLATIV 1.5 0,010 Force of reaction DPV 1.0 ANALYTICAL ABSOLUT Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

8 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 8/13 Responsable : FRNANDS Roméo Clé : V Révision : 6 Modeling D 6.1 Characteristics of modeling Modeling is ideical to modeling B with a mapping composed of triangles. Modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_DIL with the choice of ierpolation P2-P1-P Characteristics of the grid The group of meshs ROCH_RG St obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 441 Number of SG3 40 Number of TRIA6 400 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme GONF 1.0 N1 NON-RGRSSION RLATIV Displaceme DX 1.0 N5 NON-RGRSSION RLATIV Force of reaction DPV 1.0 NON-RGRSSION ABSOLU T Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

9 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 9/13 Responsable : FRNANDS Roméo Clé : V Révision : 7 Modeling 7.1 Characteristics of modeling Modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_DIL with the choice of ierpolation P2-P1-P0. The case prese is ideical to modeling A with r=0 we have then { 1 = u 1 2 u 3 = 4.u u S 1 = 3 2.a1.u Characteristics of the grid The group of meshs ROCH_RG St obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 8 Number of SG3 4 Number of QUAD8 2 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme GONF 1.0 N1 ANALYTICAL RLATIV ,010 Displaceme DX 1.0 N5 ANALYTICAL RLATIV ,010 Force of reaction SIG1 1.0 ANALYTICAL RLATIV 1.5 0,010 Force of reaction DPV 1.0 ANALYTICAL RLATIV ,010 Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

10 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 10/13 Responsable : FRNANDS Roméo Clé : V Révision : 8 Modeling F 8.1 Characteristics of modeling The case prese is ideical to modeling A with a generalization of the behavior second complete gradie [R ]. Modeling relating to the first gradie of the field of displaceme is D_PLAN_SI and that bearing on the second gradie of dilation is D_PLAN_2DG with the choice of ierpolation P2-P1-P0. We show as for modeling that: u 1 = 1 in the same way u 3 = u 1 2 For the calculation of 111 and 221 for the definition of the field of double constrais we thus have: 111 =a a a with: {a =2 a 1 a 2 a 3 a 4 a 5 a 23 =a 2 2a 3 a 12 =a 1 a2 2 Being given the values of the parameters material and limiting conditions we find: 111 =a 1.u 1 in the same way we end to: 221 = a1 2. u Characteristics of the grid The group of meshs ROCH_RG St obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second complete gradie for the regularization. Many nodes 8 Number of SG3 4 Number of QUAD8 2 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme V N1 ANALYTICAL RLATIV ,010 Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

11 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 11/13 Responsable : FRNANDS Roméo Clé : V Révision : Displaceme DX 1.0 N5 ANALYTICAL RLATIV Force of reaction SIG ANALYTICAL RLATIV Force of reaction SIG ANALYTICAL RLATIV Force of reaction DPV ANALYTICAL RLATIV , , , ,010 Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

12 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 12/13 Responsable : FRNANDS Roméo Clé : V Révision : 9 Modeling G 9.1 Characteristics of modeling Modeling G. is ideical to modeling A but with one ierpolation without multipliers of Lagrange (P2- P1) to see R Characteristics of the grid The group of meshs ROCH_RG St obtained by duplication of group of meshs named ROCH, whose objective is to accommodate modeling second gradie of dilation for the regularization. Many nodes 8 Number of SG3 4 Number of QUAD8 2 Number of group of meshs Sizes tested and results Value tested Mome Node Type Criterion Aster Tolerance Displaceme GONF 1.0 N1 NON-RGRSSION RLATIV -0, Displaceme DX 1.0 N5 NON-RGRSSION RLATIV 0, Constrai SIG1 1.0 NON-RGRSSION RLATIV 1, Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

13 Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 13/13 Responsable : FRNANDS Roméo Clé : V Révision : 10 Summary of the results This test makes it possible to check in a very simple way the good performance of the modeling second gradie, which also coincides with the results with the analytical solution. Copyright 2017 DF R&D - Licensed under the terms of the GNU FDL (

SSLL117 Validation of the modelizations second Summarized

Titre : SSLL117 - Validation des modélisations second grad[...] Date : 21/10/2011 Page : 1/13 SSLL117 Validation of the modelizations second Summarized gradient: This test makes it possible to validate