ملاحظات در مورد تجزیه و تحلیل حساسیت شکلی تغییرات بر اساس مختصات محلی

Shape and topological sensitivity analysis are two closely related research fields of both theoretical and computational mechanics with a high impact on any analytical and numerical approach in structural optimisation. There are close connections to configurational mechanics describing cracks and dislocations as well as to biomechanics observing growth and morphogenesis. Different approaches exist to compute the gradients needed by nonlinear programming algorithms. But it is of utmost importance to acknowledge that mainly a rigorous analysis of the sensitivities provides the deep insight into the nature of the mechanical problems needed to model and to solve inverse problems efficiently. This paper outlines the author's concept of an intrinsic formulation in local coordinates of continuum mechanics which extends Noll's intrinsic concept to variable material bodies. This viewpoint is derived by a thorough analysis of their manifold properties and yields the separation of the phenomena in material space from the motion in physical space. The subsequent variational shape sensitivity analysis is formulated and compared to known approaches. The interactions with computational techniques such as computer aided geometrical design (CAGD), the finite element method (FEM) and the boundary element method (BEM) are highlighted. Furthermore, the implications on the numerical algorithms for the discrete sensitivity analysis are outlined. Finally, the challenges of a singular value decomposition (SVD) of the resulting sensitivities are discussed.

Engineering analysis applied to a broad spectrum of problems from solid mechanics is based on either the finite element method (FEM), or the boundary element method (BEM) or on a combination of both. The following steps can be recognised in modelling and simulation of all mechanical phenomena. (i) Field theories are used to formulate basics of kinematics, equilibrium and material laws. (ii) Strong and weak equilibrium conditions are formulated for continuous solutions. (iii) Approximation schemes yield the algebraic equations characterising discrete solutions. (iv) Implementations of the numerical algorithms lead to executable programs. (v) Applications of these numerical schemes solve engineering problems. The known theoretical and computational approaches to shape sensitivity analysis can be linked to the different stages in the modelling and the simulation process.

The aims and objectives of the author's research are the canonical integration of shape modifications into continuum mechanics with a minimal and optimal enhancement of its theoretical foundation. Furthermore, the obtained theoretical results should be computed numerically on the same level of sophistication as it is nowadays standard in computational mechanics. This paper highlights the so far obtained results which can be summarised as follows. The starting point of any consideration should be the intrinsic formulation which is advocated by Walter Noll as an Update of The Nonlinear Field Theories of Mechanics, see the detailed arguments for this improvements in [20], [22] and [23]. The intrinsic formulation in local coordinates or the local formulation in short form has been proposed as a step towards a computable interpretation of the original intrinsic ideas. It was shown that several computational techniques, e.g. some important details of the isoparametric finite element scheme as well as the kinematical concept of the ALE approach, can be directly deduced from or replaced by the intrinsic formulation using its local interpretation. The desired enhancement for shape optimisation as well as for other related problems with non-classical phenomena such as configurational mechanics is the concept of design modifications of the reference placement mapping κκ and of the configuration GΘGΘ. Thus, continuum mechanics can be interpreted as a field theory of two fundamental mappings offering a dual treatment of direct and inverse problems. The superior intrinsic concept and especially the inherent available separation of the design dependent configuration GΘ(s)GΘ(s) and the time dependent deformation process gΘ(t)gΘ(t) offers the possibility for an integrated theoretical and algorithmic development process. Two computational aspects are highlighted in this paper. Firstly, the presented approach generates the components of the Fréchet derivatives relating the discrete space of design variations with the discrete space of the structural response. This is a major enhancement compared to traditional approaches which compute any adjoint load separately using Gâteaux derivatives. Secondly, the novel outline on the challenges of a singular value decomposition (SVD) of the Fréchet operator offers the chance to quantify engineering decision. Finally, the variational design sensitivity analysis is seen to be a prerequisite to tackle more interesting questions rather than computing the pure numbers for any gradient based nonlinear programming approach. Error analysis and postprocessing information for engineers may be challenging opportunities to highlight the advantages of variational shape sensitivity analysis.