3d Adjoint Research Articles

Topology optimization for magnetic circuits with continuous adjoint method in 3D

PurposeThe purpose of this paper is to present a new approach to optimizing the design of 3D magnetic circuits. This approach is based on topology optimization, where derivative calculations are performed using the continuous adjoint method. Thus, the continuous adjoint method for magnetostatics has to be developed in 3D and has to be combined with penalization, filtering and homotopy approaches to provide an efficient optimization code.Design/methodology/approachTo provide this new topology optimization code, this study starts from 2D magnetostatic results to perform the sensitivity analysis, and this approach is extended to 3D. From this sensitivity analysis, the continuous adjoint method is derived to compute the gradient of an objective function of a 3D topological optimization design problem. From this result, this design problem is discretized and can then be solved by finite element software. Thus, by adding the solid isotropic material with penalization (SIMP) penalization approach and developing a homotopy-based optimization algorithm, an interesting means for designing 3D magnetic circuits is provided.FindingsIn this paper, the 3D continuous adjoint method for magnetostatic problems involving an objective least-squares function is presented. Based on 2D results, new theoretical results for developing sensitivity analysis in 3D taking into account different parameters including the ferromagnetic material, the current density and the magnetization are provided. Then, by discretizing, filtering and penalizing using SIMP approaches, a topology optimization code has been derived to address only the ferromagnetic material parameters. Based on this efficient gradient computation method, a homotopy-based optimization algorithm for solving large-scale 3D design problems is developed.Originality/valueIn this paper, an approach based on topology optimization to solve 3D magnetostatic design problems when an objective least-squares function is involved is proposed. This approach is based on the continuous adjoint method derived for 3D magnetostatic design problems. The effectiveness of this topology optimization code is demonstrated by solving the design of a 3D magnetic circuit with up to 100,000 design variables.

Shape sensitivity of thermoacoustic oscillations in an annular combustor with a 3D adjoint Helmholtz solver

We model acoustic oscillations driven by velocity-coupled heat release rate fluctuations. We obtain an inhomogeneous wave equation and convert it to the frequency domain with a modal decomposition. We impose acoustic boundary conditions and, using a finite element discretization with Lagrange elements, express this as a nonlinear eigenvalue problem for the complex angular frequency, ω. We solve this using the open source platform FEniCS combined with the SLEPc, PETSc and OpenMPI libraries. In Hadamard form we write the derivative of the eigenvalue, ω, with respect to generic geometry changes. This requires the solution of the adjoint equation, which we obtain with the same method as the direct equation. The output is a thermoacoustic Helmholtz solver that cheaply calculates the effect of generic shape changes on the growth rate and frequency of thermoacoustic oscillations. We then consider symmetry-preserving and symmetry-breaking geometry modifications. For demonstration we model a 30kW laboratory-scale annular combustor (MICCA from EM2C). We parametrize the surfaces of the three-dimensional geometry with NURBS using control points. For the plenum and the combustion chamber, we find the eigenvalue shape derivatives with respect to the parameters of these control points. We apply two different strategies, perpendicular boundary movements and control point perturbations, to implement shape changes proportional to these shape derivatives, thereby reducing the growth rate of the unstable mode by increasing the phase shift between the pressure and the heat release rate oscillations. This computational method shows how to significantly alter thermoacoustic oscillations by making small geometry changes. The framework in this paper can handle arbitrarily complex three-dimensional geometries, which will be useful for the design of industrial combustion systems.

Thermoacoustic stabilization of combustors with gradient-augmented Bayesian optimization and adjoint models

Bayesian optimization (BO) is a global optimization algorithm well-suited for multimodal functions that are costly to evaluate, e.g. quantities derived from computationally expensive simulations. Recent advances have made it possible to scale BO to high-dimensional functions and accelerate its convergence by incorporating derivative information. These developments have laid the groundwork for a productive interplay between BO and adjoint solvers, a tool to cheaply obtain gradients of objective functions w.r.t. tunable parameters in a simulated physical system. In thermoacoustics, adjoint-based optimization has previously been applied to Helmholtz solvers and low-order network models to find optimally stable combustor configurations. These studies have used conjugate gradient or quasi-Newton optimizers which can get stuck in local optima and may require many evaluations of the underlying model to find a good optimum. In this paper, we propose using gradient-augmented BO to optimize adjoint models. We consider two test cases from the thermoacoustics literature: optimizing design parameters in a 1D adjoint Helmholtz model of a Rijke tube and geometry optimization in a low-order network model of a longitudinal combustor. We show that compared to BFGS, a standard quasi-Newton method, our gradient-enhanced BO arrives at multiple, more optimal configurations using considerably fewer evaluations of the solver. This approach holds great promise for efficient thermoacoustic stabilization when designing using expensive 3D adjoint Helmholtz solvers.

Advanced two-level CMFD acceleration method for the 3D whole-core high-fidelity neutron adjoint transport calculation

In the 2D/1D method, a global adjoint CMFD based on the generalized equivalence theory is built to synthesize the 2D radial MOC adjoint and 1D axial NEM adjoint calculation and also to accelerate the iteration convergence of 3D whole-core adjoint transport calculation. Even more important, an advanced yet accurate two-level (TL) CMFD acceleration technique is proposed, in which an equivalent one-group adjoint CMFD is established to accelerate the multi-group adjoint CMFD and then to accelerate the 3D whole-core adjoint transport calculation efficiently. Based on these method, a new code is developed to perform 3D adjoint neutron flux calculation. Then a set of VERA and C5G7 benchmark problems are chosen to verify the capability of the 3D adjoint calculations and the effectiveness of TL CMFD acceleration. The numerical results demonstrate that acceptable accuracy of 2D/1D adjoint calculations and superior acceleration of TL CMFD are achievable.

Screening in hot SU(2) gauge theory and propagators in 3d adjoint Higgs model

We investigate the large distance behavior of the electric and magnetic propagators of hot SU(2) gauge theory in different gauges using lattice simulations of the full 4d theory and the effective, dimensionally reduced 3d theory. A comparison of the 3d and 4d data for the propagators suggests that dimensional reduction works surprisingly well down to temperatures T ∼ 2 T c . A detailed study of the volume dependence of magnetic propagators is performed. The electric propagators show exponential decay at large distances in all gauges considered and a possible gauge dependence of the electric screening mass turns out to be statistically insignificant.

Propagators of hot SU(2) gauge theory from 3d adjoint Higgs model

We study propagators of the lattice 3d adjoint Higgs model, considered as an effective theory of 4d SU(2) gauge theory at high temperature. The propagators are calculated in so-called λ-gauge. ¿From the long distance behaviour of the propagators we extract the screening masses. It is shown that the pole masses extracted from the propagators agree well with the screening masses obtained recently in finite temperature SU(2) theory. The gauge dependence of the screening masses is also discussed.

Propagators of hot SU(2) gauge theory from 3d adjoint Higgs model

Propagators of hot SU(2) gauge theory from 3d adjoint Higgs model

Screening masses of hot SU(2) gauge theory from the 3d adjoint Higgs model

We study the Landau gauge propagators of the lattice SU(2) 3d adjoint Higgs model, considered as an effective theory of high temperature 4d SU(2) gauge theory. From the long distance behaviour of the propagators we extract the screening masses. The propagators are studied both in the symmetric and the broken phases of the 3d Higgs model. It is shown that the pole masses extracted from the propagators in the symmetric phase agree well with the screening masses obtained recently in finite temperature SU(2) theory, while propagators measured in the broken phase show quite a different behaviour. This suggest that the symmetric phase of the 3d model corresponds to the deconfined phase of the 4d SU(2) gauge theory. The relation of the propagator masses to the masses extracted from gauge invariant correlators and the mass gap of pure 3d SU(2) gauge theory is also discussed.

ASYNT — An adjoint synthesis method for neutron irradiation estimation on the vessels of VVER/PWR reactors

The new ASYNT method is developed and proposed for neutron fluence calculations. This method uses the solution of adjoint neutron transport equation for flux/responses evaluation. The evaluation of flux/responses is reduced to the space and energy integration of the product of 3D adjoint solution and the neutron source distribution, determined by realized loading patterns and operational regimes. The adjoint solution does not depend on the neutron source distribution and is obtained only once for every surveillance site, response and type of reactor. The application of this method results in separability of azimuthal and axial dependence in the 3D adjoint solution. That is why the 3D adjoint solution could be synthesised from the 2D and 1D adjoint solutions. The circular cylinder reactor core presentation of the solution axial dependence is the only approximation used in the ASYNT method.