Discrete adjoint methodology for general multiphysics problems

This article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account.

Isogeometric shape optimization for design dependent loads

This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behaviour of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using NURBS and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with CAD. Two classes of optimization problems have been discussed, they are minimum compliance problems subject to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which is new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Application of the Complex Variable Step Method and the Boundary Element Method for Sensitivity Analysis of Steady Heat Conduction Problems

The paper concerns the problems related to applying the complex variable step method for the sensitivity analysis of the steady temperature field in the solid body domain due to the perturbations of the geometrical and physical parameters. The optimization problem using the approach proposed is also discussed. At the stage of numerical modelling, the boundary element method is used. The first part of the paper is devoted to the shape sensitivity. The results obtained are compared with the solution resulting from the implicit approach of sensitivity analysis. In the second part, the practical problem concerning optimizing the geometry of continuous casting mould cross-section is considered. The project variable vector contains the cooling pipes' radius and the volume flow rate of the cooling water. The numerical results and the conclusions are presented in the final part of the paper.

Two-scale shape optimisation based on numerical homogenisation techniques and variational sensitivity analysis

This contribution presents a theoretical and computational framework for two-scale shape optimisation of nonlinear elastic structures. Particularly, minimum compliance optimisation problems with composite (matrix-inclusion) microstructures subjected to static loads and volume-type design constraints are focused. A homogenisation-based FE$$^2$$ 2 scheme is extended by an enhanced formulation of variational (shape) sensitivity analysis based on Noll’s intrinsic, frame-free formulation of continuum mechanics. The obtained overall two-scale sensitivity information couples shape variations across micro- and macroscopic scales. A numerical example demonstrates the capabilities of the proposed variational sensitivity analysis and the (shape) optimisation framework. The investigations involve a mesh morphing scheme for the design parametrisation at both macro- and microscopic scales.