A New Lighting on Analytical Discrete Sensitivities in the Context of IsoGeometric Shape Optimization

Abstract

Isogeometric shape optimization has been now studied for over a decade. This contribution aims at compiling the key ingredients within this promising framework, with a particular attention to sensitivity analysis. Based on all the researches related to isogeometric shape optimization, we present a global overview of the process which has emerged. The principal feature is the use of two refinement levels of the same geometry: a coarse level where the shape updates are imposed and a fine level where the analysis is performed. We explain how these two models interact during the optimization, and especially during the sensitivity analysis. We present new theoretical developments, algorithms, and quantitative results regarding the analytical calculation of discrete adjoint-based sensitivities. In order to highlight the versatility of this sensitivity analysis method, we perform eight benchmark optimization examples with different types of objective functions (compliance, displacement field, stress field, and natural frequencies), different types of isogeometric element (2D and 3D standard solids, and a Kirchhoff-Love shell), and different types of structural analysis (static and vibration). The numerical performances of the analytical sensitivities are compared with approximate sensitivities. The results in terms of accuracy and numerical cost make us believe that the presented method is a viable strategy to build a robust framework for shape optimization.

Fast and multiscale formation of isogeometric matrices of microstructured geometric models

“…Except this adjoint field, the differentiation of the Jacobian determinant w.r.t. the control point coordinates in Equation ( 66 ) is a standard mathematical operation found also in the classical isogeometric shape optimization framework, see for example [ 38 ] for the calculation details. Finally, the computational cost of the gradient as given by Equation ( 66 ) is almost the same than computing the gradient associated to the macro-geometry without the microstructure.…”

“…Finally, the derivatives of the macro-fields w.r.t. the control points in Equation ( 70 ) are standard calculation steps in structural shape optimization, see for example [ 38 , 60 ] for the details. The differentiation of the external work in Equation ( 69 ) is done similarly.…”

Quadrature-free Immersed Isogeometric Analysis

“…Finally, in Section 5.2.3, the flexibility and robustness of the proposed approach is demonstrated in the case of geometries that present a level complexity analogous to the ones found in real industrial applications. For all the studied cases, we consider the approximated Poisson's problem (11), previously discussed in Section 2. We adopt manufactured solutions:…”

“…Let us now study the involved polynomial degrees for the three examples included in this section according to the estimation detailed in Section 4.3. Applying the quadrature-free approach to solve the Poisson's problem (11), we can identify the polynomial integrand a (recall Equation ( 26)) with the term 18), where we assumed K to be the identity and therefore the projection degrees to be q = (0, 0, 0)).…”

“…Particu-P. Antolin -T. Hirschler Institute of Mathematics, Chair of Numerical Modelling and Simulation École Polytechnique Fédérale de Lausanne, Switzerland Email: pablo.antolin@epfl.ch larly, spline-based geometric models have been found to present excellent performance for numerical simulations [4][5][6][7]. This opens the door to the formation of all-in-one design frameworks where a single geometric model is simultaneously used for parameterizing the shape of the object of interest and performing advanced numerical analyses [8][9][10][11]. The combination into one single model of both high-fidelity geometrical properties and efficient analysis performances is however far from trivial in general.…”