Optimized lattice-based metamaterials for elastostatic cloaking

An optimization-based approach is proposed to design elastostatic cloaking devices in two-dimensional (2D) lattices. Given an elastic lattice with a defect, i.e. a circular or elliptical hole, a small region (cloak) around the hole is designed to hide the effect of the hole on the elastostatic response of the lattice. Inspired by the direct lattice transformation approach to elastostatic cloaking in 2D lattices, the lattice nodal positions in the design region are obtained using a coordinate transformation of the reference (undisturbed) lattice nodes. Subsequently, additional connectivity (i.e. a ground structure) is defined in the design region and the stiffness properties of these elements are optimized to mimic the global stiffness characteristics of the reference lattice. A weighted least-squares objective function is proposed, where the weights have a physical interpretation—they are the design-dependent coefficients of the design lattice stiffness matrix. The formulation leads to a convex objective function that does not require a solution to an additional adjoint system. Optimization-based cloaks are designed considering uniaxial tension in multiple directions and are shown to exhibit approximate elastostatic cloaking, not only when subjected to the boundary conditions they were designed for but also for uniaxial tension in directions not used in design and for shear loading.

A Simple Approach for the Computation of Lyapunov-Floquet Transformations for the Mathieu Equation

Lyapunov-Floquet (L-F) transformations reduce linear ordinary differential equations with time-periodic coefficients (so-called linear time-periodic systems) to equations with constant coefficients. The present work proposes a simple approach to construct L-F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential term and a periodic term. Using this Floquet form of a solution, the ordinary differential equation corresponding to a linear time-periodic system reduces to an eigenvalue problem. Next, eigenanalysis is performed to obtain the general solution and subsequently, the state transition matrix of the time-periodic system is constructed. Then, the Lyapunov-Floquet theorem is used to compute L-F transformation. The inverse of L-F transformation is determined by defining the adjoint system to the time-periodic system. Mathieu equation is investigated in this work and L-F transformations and their inverse are generated for stable and unstable cases. These transformations are very useful in the design of controllers using time-invariant methods and in the bifurcation studies of nonlinear time-periodic systems.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

An optimal control method for time-dependent fluid-structure interaction problems

In this article, we derive an adjoint fluid-structure interaction (FSI) system in an arbitrary Lagrangian-Eulerian (ALE) framework, based upon a one-field finite element method. A key feature of this approach is that the interface condition is automatically satisfied and the problem size is reduced since we only solve for one velocity field for both the primary and adjoint system. A velocity (and/or displacement)-matching optimisation problem is considered by controlling a distributed force. The optimisation problem is solved using a gradient descent method, and a stabilised Barzilai-Borwein method is adopted to accelerate the convergence, which does not need additional evaluations of the objective functional. The proposed control method is validated and assessed against a series of static and dynamic benchmark FSI problems, before being applied successfully to solve a highly challenging FSI control problem.

Robust Assignment of Natural Frequencies and Antiresonances in Vibrating Systems through Dynamic Structural Modification

This paper proposes a novel method for the robust partial assignment of natural frequencies and antiresonances, together with the partial assignment of the related eigenvectors, in lightly damped linear vibrating systems. Dynamic structural modification is exploited to assign the eigenvalues, either of the system or of the adjoint system, together with their sensitivity with respect to some parameters of interest. To handle with constraints on the feasible modifications, the inverse eigenvalue problem is cast as a minimization problem and a solution method is proposed through homotopy optimization. Variables lifting for bilinear and trilinear terms, together with bilinear and double-McCormick’s constraints, are exploited to provide a convexification of the problem and to boost the attainment of the global optimum. The effectiveness of the proposed method is assessed through four numerical examples.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Local null controllability of a model system for strong interaction between internal solitary waves

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

Adjoint-based exact Hessian computation

We consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role.

Nonlinear Schrodinger Equation-Based Adjoint Sensitivity Analysis

A general nonlinear adjoint sensitivity analysis (ASA) approach for the time-dependent nonlinear Schrodinger equation (NLSE) is presented. The proposed algorithm estimates the sensitivities of a desired objective function with respect to all design parameters using only one extra adjoint system simulation. The approach efficiency is shown here through a numerical example.

Null controllability for a class of stochastic singular parabolic equations with the convection term

<p style='text-indent:20px;'>This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.</p>