Adjoint System Method Research Articles

CFD-based geometrical shape optimization of a packed-bed reactor combining multi-objective and adjoint system methods

This paper presents the development of a geometric shape optimization methodology based on the so-called ”Hadamard boundary variation” method for performing very general domain deformations, and the related concept of domain differentiation. The resulting method is used to determine the optimal configuration of a two-dimensional packed-bed reactor that simultaneously optimizes its conversion rate and fluid energy dissipation, and where a homogeneous first-order reaction or a catalytic surface reaction takes place. The considered multi-objective optimization problem is subjected to four constraints: the process model constraints consisting of the Navier–Stokes, continuity and mass balance equations, an iso-volume and two manufacturing constraints. The approach to solve the problem is based on the linear scalarization method which converts the multi-objective problem into a single objective problem. The adjoint system method is used to compute the gradient of the performance indices and constraints. Since the indices are conflicting, the solution of the problem is a set of solutions, called Pareto front. Each optimal solution is evaluated using multi-attribute utility theory (MAUT) to determine the best optimal shape of the reactor. Finally, the resulting shape is fabricated using a 3D printing technique and will be experimentally validated.

Optimal design of a structured fixed-bed reactor using geometry optimization and Stratoconception printing process

The aim of this paper is to determine the shape of a fixed-bed reactor which maximizes the conversion rate under the constraints of process model equations (i.e. continuity, Navier–Stokes, and mass balance equations), energy dissipation, iso-volume, and manufacturing. Incompressible fluid, laminar flow regime and steady-state conditions in the reactor are the main assumptions taken into account. The optimization method developed is based on the adjoint system method and OpenFOAM framework is used as CFD solver to compute the state vector and its adjoint variables introduced by the optimization approach. The algorithm developed is then tested on two different cases, a reactor where a first order homogeneous reaction takes place and another one involving a surface reaction. The optimization results show a significant improvement of the conversion rate by 2.7% in the first case, and by 16% in the second one. Finally, initial and optimal shapes are manufactured using a 3D printing technique.

Shape Optimization of Fixed-Bed Reactors in Process Engineering

This paper deals with geometric shape optimization of a parallelepiped shaped fixed-bed reactor with single phase liquid flow where a chemical reaction takes place. The packing is a sort of static mixer made up of solid cylindrical obstacles uniformly distributed in the reactor. The reactive flow is described by means of momentum and mass transport equations in laminar flow regime. The objective is to determine the shape of the packing that maximizes the reaction conversion rate subjected to some specified manufacturing constraints. The optimization approach developed is based on the adjoint system method and the results show that the optimal shape obtained allows one to significantly improve the reaction conversion rate. Furthermore, the optimal shape is printed by means of an additive manufacturing technique and several manufacturing constraints mainly related to the thickness of packing are considered.

Time-independent stochastic design sensitivity analysis of structural systems with second-order accuracy

In the paper, the static sensitivity of complex structures with respect to random design parameters is presented. Using the adjoint system method, based on the mean-point second-order perturbation method, the first two probabilistic moments of time-independent sensitivity are formulated with means and cross-covariances of random design parameters as the input data. It enables one to obtain the second-order accuracy of the solution. The presented formulations are illustrated by a number of numerical examples. The influence of finite element mesh density for the obtained results is discussed using the analysis of the spatial bar dome.

Error variance-constrained ℋ∞ filtering for a class of nonlinear stochastic systems with degraded measurements: the finite horizon case

This article is concerned with the robust ℋ∞ filtering problem for a class of time-varying nonlinear stochastic systems with error variance constraint. The stochastic nonlinearities considered are quite general, which contain several well-studied stochastic nonlinear systems as special cases. The purpose of the filtering problem is to design a filter which is capable of achieving the pre-specified ℋ∞ performance and meanwhile guaranteeing a minimised upper-bounded on the filtering error variance. By means of the adjoint system method, a necessary and sufficient condition for satisfying the ℋ∞ constraint is first given, expressed as a forward Riccati-like difference equation. Then an upper-bound on the variance of filtering error system is given, guaranteeing the error variance is not more than a certain value at each sampling instant. The existence condition for the desired filter is established, in terms of the feasibility of a set of difference Riccati-like equations, which can be solved forward in time, hence is suitable for online computation. A numerical example is presented finally to show the effectiveness and applicability of the proposed method.

Multi-resolution based sensitivity analysis of complex non-linear circuits

This study addresses the sensitivity analysis of non-linear circuits in their transient and periodic behaviour. The circuits here considered are built of connections of N-ports with frequency-dependent parameters whose input and output quantities are expanded in the wavelet domain. The use of wavelet instead of the Fourier analysis allows a significant increase in the sparsity of the matrices with a comparable accuracy and convergence rate. In addition, by using the proper wavelet basis, it is possible to analyse both the steady state and the transient behaviour. The adjoint system method is used to obtain the sensitivities of the response of the circuit with respect to the design parameters. The multiport connection is described in terms of scattering parameters and the hierarchical approach is extended to the adjoint system. Computational aspects are discussed and examples of application of the proposed technique are reported. The results are compared with those obtained by the use of other techniques.

Efficient Sensitivity Calculations for Optimization of Power Delivery Network Impedance

This paper presents efficient algorithms for sensitivity calculations of power delivery network (PDN) impedance. The sensitivity algorithm is based on the adjoint system method. Efficient calculation of sensitivity is critical for optimization of complex PDNs. The frequency-domain design of a PDN is a multiobjective optimization problem, which is formulated as a minimax problem in this paper. The design variables are typically the decoupling capacitors (decaps). The objective function is the input impedance of the PDN calculated at relevant frequencies. This paper also presents an efficient algorithm for large-change sensitivity calculations. Based on this method, what-if scenarios can be simulated efficiently, such as the addition, modification, or relocation of a few decaps or a small change in the geometry.

ROBUST MEASUREMENT-BASED OPTIMIZATION WITH STEADY-STATE DIFFERENTIAL EQUATION MODELS

A robust model-based iterative feedback optimization methodolgy has been introduced in previous work for the steady-state optimization of chemical process operations without requiring cumbersome model updating. Here we extend the methodology to utilize differential equation models directly. The adjoint system method that is often used in dynamic optimization is modified to explicitly incorporate plant measurements in the gradient computation during each iteration, resulting in robustness with respect to model-plant mismatch. In the case that the states cannot be measured along the entire spatial direction, estimated profiles based on boundary measurements are utilized. The methodology is tested with simulations of an ammonia synthesis reactor in autothermal operation and is shown to be robust in the presence of modeling error, input error, and measurement noise.

Parameter sensitivity of elastoplastic response

The general problem of sizing, material and loading parameter sensitivity of non‐linear systems is presented. Both kinematic and path‐dependent material non‐linearities are considered; non‐linear sensitivity path is traced by an incremental solution strategy. The variational approach employed is quite general and can be employed for studying sensitivity of various path‐dependent highly non‐linear phenomena. Both the direct differentiation method (DDM) and adjoint system method (ASM) are discussed in the context of continuum and finite element mechanics. The merits of using the consistent tangent matrix and the necessity of accumulation of design derivatives of stresses and internal parameters are indicated. Aspects of sensitivity problems in metal forming are also discussed. A number of examples illustrate the paper.

Boundary elements in shape design sensitivity analysis and optimal design of vibrating structures

A general approach to shape design sensitivity analysis and optimal design for dynamic transient and free vibration problems using boundary elements is presented. The material derivatives and the adjoint system method are applied to obtain first-order sensitivities for the effect of boundary shape variations. A numerical example of shape sensitivity analysis and optimal design for free vibrations of an elastic body is presented.

Boundary elements in shape design sensitivity analysis and optimal design of vibrating structures

A general approach to shape design sensitivity analysis and optimal design for dynamic transient and free vibrations problems using boundary elements is presented. The material derivatives and the adjoint system method are applied to obtain first-order sensitivities for the effect of boundary shape variations. A numerical example of shape sensitivity analysis and optimal design for free vibrations of an elastic body is presented.