Adjoint-based methods to compute higher-order topological derivatives with an application to elasticity

PurposeThe goal of this paper is to give a comprehensive and short review on how to compute the first- and second-order topological derivatives and potentially higher-order topological derivatives for partial differential equation (PDE) constrained shape functionals.Design/methodology/approachThe authors employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint-based methods to compute higher-order topological derivatives. To illustrate the methodology proposed in this paper, the authors then apply the methods to a linear elasticity model.FindingsThe authors compute the first- and second-order topological derivatives of the linear elasticity model for various shape functionals in dimension two and three using Amstutz' method, the averaged adjoint method and Delfour's method.Originality/valueIn contrast to other contributions regarding this subject, the authors not only compute the first- and second-order topological derivatives, but additionally give some insight on various methods and compare their applicability and efficiency with respect to the underlying problem formulation.

On design sensitivities in the structural analysis and optimization of flexible multibody systems

The structural analysis and optimization of flexible multibody systems become more and more popular due to the ability to efficiently compute gradients using sophisticated approaches such as the adjoint variable method and the adoption of powerful methods from static structural optimization. To drive the improvement of the optimization process, this work addresses the computation of design sensitivities for multibody systems with arbitrarily parameterized rigid and flexible bodies that are modeled using the floating frame of reference formulation. It is shown that it is useful to augment the body describing standard input data files by their design derivatives. In this way, a clear separation can be achieved between the body modeling and parameterization and the system simulation and analysis.

Acoustic Shape Optimization Based on Isogeometric Wideband Fast Multipole Boundary Element Method with Adjoint Variable Method

A shape optimization approach based on isogeometric wideband fast multipole boundary element method (IGA WFMBEM) in 2D acoustics is developed in this study. The key treatment is shape sensitivity analysis by using the adjoint variable method under isogeometric analysis (IGA) conditions. A set of efficient parameters of the wideband fast multipole method has been identified for IGA boundary element method. Shape optimization is performed by applying the method of moving asymptotes. IGA WFMBEM is validated through an acoustic scattering example. The proposed optimization approach is tested on a sound barrier and two multiple structures to demonstrate its potential for engineering problems.

Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

An optimal bounded control strategy for strongly nonlinear vibro-impact systems under stochastic excitations with actuator saturation is proposed. First, the impact effect is incorporated in an equivalent equation by using a nonsmooth transformation. Under the assumption of light damping and weak random perturbation, the system energy is a slowly varying process. By using the stochastic averaging of envelope for strongly nonlinear systems, the partially averaged Itô stochastic differential equation for system energy can be derived. The optimal control problem is transformed from the original optimal control problem for the state variables to an equivalent optimal control problem for the system energy, which decreases the dimensions of the optimal control problem. Then, based on stochastic maximum principle, an adjoint equation for the adjoint variable and the maximum condition of partially averaged control problem are established. For infinite time-interval ergodic control, the adjoint variable is assumed to be a stationary process and the adjoint equation can be further simplified. Finally, the probability density function of the system energy and other statistics of the optimally controlled system are derived by calculating the associated Fokker–Plank–Kolmogorov equation. For comparison, the bang–bang control is also investigated and the control results are compared to show the advantages of the developed control strategy.