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**Shape design sensitivities improvement using convected unstructured meshes.**
*(English)*
Zbl 1293.74347

Summary: This paper focuses on various forms of direct differentiation methods for design sensitivity computation in the shape optimisation of continuum structures and the role of convected meshes on the accuracy of the sensitivities. A Pseudo-Analytical Sensitivity Analysis (P-ASA) method is presented and tested. In this method the response analysis component uses unstructured finite element meshes and the sensitivity algorithm entails shape-perturbation for each design variable. A material point is convected during a change of shape and the design sensitivities are therefore intrinsically associated with the mesh-sensitivities of the finite element discretization. Such mesh sensitivities are obtained using a very efficient boundary element point-tracking analysis of an affine notional underlying elastic domain. All of the differentiation, with respect to shape variables, is done exactly except for the case of mesh-sensitivities: hence the method is almost analytical. In contrast to many other competing methods, the P-ASA method is, by definition, independent of perturbation step-size, making it particularly robust. Furthermore, the sensitivity accuracy improves with mesh refinement. The boundary element point-tracking method is also combined with two popular methods of sensitivity computation, namely the global finite difference method and the semi-analytical method. Increases in accuracy and perturbation range are observed for both methods.

### MSC:

74P10 | Optimization of other properties in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

74S15 | Boundary element methods applied to problems in solid mechanics |

### Keywords:

shape optimization; sensitivity analysis; analytical gradients; convected unstructured FE meshes
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\textit{D. Kelliher} and \textit{J. S. Campbell}, Optim. Eng. 14, No. 1, 175--191 (2013; Zbl 1293.74347)

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### References:

[1] | Akbari, J; Kim, NH; Ahmadi, MT, Shape sensitivity analysis with design-dependent loadings equivalence between continuum and discrete derivatives, Struct Multidiscip Optim, 40, 353-364, (2010) · Zbl 1274.74303 |

[2] | Anderson, WK; Newman, JC; Whitfield, DL; Nielsen, EJ, Sensitivity analysis of Navier-Stokes equations on unstructured meshes using complex variables, AIAA J, 39, 56-63, (2001) |

[3] | Barthelemy, B; Haftka, RT, Accuracy analysis of the semi-analytical method for shape sensitivity calculation, Mech Struct Mach, 18, 407-432, (1990) |

[4] | Barthelemy, B; Chon, CT; Haftka, RT, Accuracy problems associated with semi-analytical derivatives of static responses, Finite Elem Anal Des, 4, 249-265, (1988) · Zbl 0671.73070 |

[5] | Batina, JT, Unsteady Euler airfoil solutions using unstructured dynamic meshes, Reno, Nevada, USA |

[6] | Bletzinger, K-U; Firl, M; Daoud, F, Approximation of derivatives in semi-analytical structural optimization, Comput Struct, 86, 1404-1416, (2007) |

[7] | Borggaard, J; Burns, J, A PDE sensitivity equation method for optimal aerodynamic design, J Comput Phys, 136, 367-384, (1997) · Zbl 0903.76064 |

[8] | Borggaard J, Burns J, Cliff E, Schreck S (1998) Computational methods for optimal design and control. Birkhäuser, Basel |

[9] | Brockman, RA, Geometric sensitivity analysis with isoparametric finite elements, Commun Appl Numer Methods, 3, 495-499, (1987) · Zbl 0623.73081 |

[10] | Campbell, JS; Kelliher, D, Stuctural shape optimisation of elastic continua: a study on nonlinearity and a 2d benchmark, Swansea, July 2000 |

[11] | Cheng, G; Olhoff, N, Rigid body motion test against error in semi-analytical sensitivity analysis, Comput Struct, 46, 515-527, (1993) · Zbl 0769.73075 |

[12] | Cheng, G; Gu, Y; Zhou, Y, Accuracy of semi-analytic sensitivity analysis, Finite Elem Anal Des, 6, 113-128, (1989) · Zbl 0694.73038 |

[13] | Boer, H; Keulen, F, Error analysis of refined semi-analytical design sensitivities, Struct Optim, 14, 242-247, (1997) |

[14] | Boer, H; Keulen, F; Vervenne, K, Refined second order semi-analytical design sensitivities, Int J Numer Methods Eng, 55, 1033-1051, (2002) · Zbl 1033.74037 |

[15] | Etienne, S; Pelletier, D, A general approach to sensitivity analysis of fluid structure interactions, J Fluids Struct, 21, 169-186, (2005) |

[16] | Farhat, C; Degand, C; Koobus, B; Lesoinne, M, Torsional springs for two-dimensional dynamic unstructured fluid meshes, Comput Methods Appl Mech Eng, 163, 231-245, (1998) · Zbl 0961.76070 |

[17] | Haftka, RT; Adelman, HM, Recent developments in structural sensitivity analysis, Struct Optim, 1, 137-151, (1989) |

[18] | Hansen, JS; Liu, ZS; Olhoff, N, Shape sensitivity analysis using a fixed basis function finite element approach, Struct Multidiscip Optim, 21, 177-195, (2001) |

[19] | Jin W, Dennis BH, Wang BP (2010) Improved sensitivity analysis using a complex variable semi-analytical method. Struct Multidiscip Optim 433-439 · Zbl 0623.73081 |

[20] | Kelliher D (1999) An adaptive finite element method for optimal shape design of elastic continua. PhD thesis, Dept. of Civil & Environmental Engineering, National University of Ireland, Cork |

[21] | Olhoff, N; Rasmussen, J, Study of inaccuracy in semi-analytical sensitivity analysis—a model problem, Struct Optim, 3, 203-213, (1991) |

[22] | Olhoff, N; Rasmussen, J; Lund, E, A method of “exact” numerical differentiation for error elimination in finite-element-based semi-analytical shape sensitivity analysis, Mech Struct Mach, 21, 1-66, (1993) |

[23] | Parente, E; Sousa, JBM, Design sensitivity analysis of nonlinear structures subject to thermal loads, Comput Struct, 86, 1369-1384, (2008) |

[24] | Parente, E; Vaz, LE, Improvement of semi-analytical design sensitivities of non-linear structures using equilibrium relations, Int J Numer Methods Eng, 50, 2127-2142, (2001) · Zbl 0977.74048 |

[25] | Pedersen, P; Cheng, G; Rasmussen, J, On accuracy problems for semi-analytical sensitivity analyses, Mech Struct Mach, 17, 373-384, (1989) |

[26] | Robinson, JC; Campbell, JS; Kelliher, D, Interior point tracking in shape evolving unstructured finite element meshes, Eng Comput, 15, 721-731, (1998) · Zbl 0943.74069 |

[27] | Sackinger, PA; Schunk, PR; Rao, RR, A Newton-raphson pseudo-solid domain mapping technique for free and moving boundary problems: a finite element implementation, J Comput Phys, 125, 83-103, (1996) · Zbl 0853.65138 |

[28] | Keulen, F; Boer, H, Rigorous improvement of semi-analytical design sensitivities by exact differentiation of rigid body motions, Int J Numer Methods Eng, 42, 71-91, (1998) · Zbl 0916.73066 |

[29] | Keulen, F; Haftka, RT; Kim, NH, Review of options for structural design sensitivity analysis. part 1—linear systems, Comput Methods Appl Mech Eng, 194, 3213-3243, (2005) · Zbl 1091.74040 |

[30] | Yamazaki, K; Vanderplaats, GN, Design sensitivity analysis with isoparametric shell elements, Struct Optim, 5, 152-158, (1993) |

[31] | Yao, T-M; Choi, KK, 3-d shape optimal design and automatic finite element regridding, Int J Numer Methods Eng, 28, 369-384, (1989) · Zbl 0669.73068 |

[32] | Zienkiewicz, OC; Campbell, JS; Gallagher, RH (ed.); Zienkiewicz, OC (ed.), Shape optimisation and sequential linear programming, 109-126, (1973), New York |

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