Goal-oriented Mesh Adaptation Research Articles

SuperAdjoint: Super-resolution neural networks in adjoint-based error estimation

Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need to compute a higher resolution adjoint solution. Here, it is proposed to use super-resolution neural networks to super-resolve a fine adjoint solution from a lower-cost coarse adjoint solution: a superAdjoint. The method is compared to reference error estimators on an unsteady Burgers’ equation using the method of manufactured solutions. Two forms of the superAdjoint were implemented, a twice and a four times refining super-resolution neural network. These were used to demonstrate both the computational cost reduction and the potential for the reduction of the storage footprint of the primal problem. The first, referred to as 2×CNN, was able to reconstruct the spatially enriched adjoint solution, thus providing a robust and inexpensive local output error. The second, the 4×CNN, was able to demonstrate the reconstruction ability of super-resolution neural networks for higher upscaling factors. This was leveraged in order to subsample the primal solution in space, thus reducing substantially the storage footprint of the discrete primal solution. Both superAdjoints could achieve the desired level of accuracy when compared to the reconstruction of the refined adjoint-based error estimate. Moreover, superAdjoint was shown to be able to generalise to a new QoI that was untrained for. This gives great confidence in the use of super-resolution neural networks for the reduction of both computational cost and storage requirements of adjoint-based error estimation, and goal-oriented mesh adaptation.

Tidal turbine array modelling using goal-oriented mesh adaptation

To examine the accuracy and sensitivity of tidal array performance assessment by numerical techniques applying goal-oriented mesh adaptation. The goal-oriented framework is designed to give rise to adaptive meshes upon which a given diagnostic quantity of interest (QoI) can be accurately captured, whilst maintaining a low overall computational cost. We seek to improve the accuracy of the discontinuous Galerkin method applied to a depth-averaged shallow water model of a tidal energy farm, where turbines are represented using a drag parametrisation and the energy output is specified as the QoI. Two goal-oriented adaptation strategies are considered, which give rise to meshes with isotropic and anisotropic elements. We present both fixed mesh and goal-oriented adaptive mesh simulations for an established test case involving an idealised tidal turbine array positioned in a channel. With both the fixed meshes and the goal-oriented methodologies, we reproduce results from the literature which demonstrate how a staggered array configuration extracts more energy than an aligned array. We also make detailed qualitative and quantitative comparisons between the fixed mesh and adaptive outputs. The proposed goal-oriented mesh adaptation strategies are validated for the purposes of tidal energy resource assessment. Using only a tenth of the number of degrees of freedom as a high-resolution fixed mesh benchmark and lower overall runtime, they are shown to enable energy output differences smaller than 2% for a tidal array test case with aligned rows of turbines and less than 10% for a staggered array configuration.

Goal-Oriented Mesh Adaptivity of Three-Dimensional Neutron Transport Calculation Using Weighted Difference Scheme and Dual-Weighted Residual Error Indicators

Maintaining a reasonable balance between computational accuracy and overhead is important for neutron transport simulations of engineering problems. This paper presents a goal-oriented mesh adaptive algorithm applied to the multigroup discrete ordinates equation for fixed source and criticality problems. The dual-weighted residual (DWR) approach estimates numerical solution errors and drives local mesh refinement for specific targets, such as detector response, integral flux, and multiplication factor. We employ a reconstruction method to evaluate the spatial residuals of the fluxes obtained by the weighted difference scheme. To improve the performance of adaptive algorithms, new estimation models are proposed for adjoint fluxes needed by the DWR theory, including a regional goal model for fixed source problems and an inconsistent fission source model for k-eigenvalue problems. Additionally, we analyze the impact of the truncation of flux reconstruction and isotropic approximation of adjoint fluxes on grid error indicators and adaptive calculations. Numerical results demonstrate that for the quantities of interest, our adaptive approach saves more than 70% of computational effort and run time when obtaining a level of high accuracy comparable to that of uniform fine grids.

Multiscale analysis of composite structures with goal-oriented mesh adaptivity and reduced order homogenization

We present a multiscale finite element approach for composite structure analysis, applying reduced order homogenization to obtain an accurate surrogate model for microscale RVE computations and establishing macroscale mesh adaptivity towards an error control of a user-defined quantity of interest. By means of the so-called nonuniform transformation field analysis, reduced variables are deduced from a space–time decomposition of inelastic strain fields. Considering the macroscopic dissipation power as a volume-averaged microscopic dissipation power, closed-form macroscale constitutive relations are established, thus resulting into a reduced order homogenization problem. For ease of error estimate, we propose a multifield formulation for the reduced order problem. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, leading to error representations aiming at the quantity of interest. Accordingly, an efficient error estimator combined with a patch recovery technique avoiding nonlinear computations is proposed. The effectiveness of the resulting adaptive algorithm is illustrated by several numerical examples w.r.t. a prototype model.

Goal-Oriented Mesh Adaptation for High-Fidelity Aeroelastic Simulations

Goal-Oriented Mesh Adaptation for High-Fidelity Aeroelastic Simulations

Goal-oriented mesh adaptivity for inverse problems in linear micromorphic elasticity

In this work, we extend goal-oriented mesh adaptivity to parameter identification for a class of linear micromorphic elasticity problems. Starting from a compact formulation in our previous work (Ju and Mahnkhen, 2017), we propose a two-level optimization framework based on goal-oriented error estimation. By means of a sensitivity analysis of the generalized constitutive relations, we establish a gradient-based solver for the inverse problem, where parameters are optimized within an inner optimization loop for a given mesh. Exact error representations are derived from a Lagrange method, aiming at a quantity of interest as a user-defined functional of the parameters. By using a patch recovery technique for enhanced solutions, a computable error estimator is presented and used to drive an adaptive refinement algorithm, which forms an outer optimization loop. For a numerical study, we investigate the performance of the resulting adaptive procedure in case of perfect, incomplete and perturbed data. The results confirm the effectiveness of the proposed adaptive procedure.

Goal-oriented mesh adaptation method for nonlinear problems including algebraic errors

We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances the errors arising from the discretization and the use of nonlinear as well as linear iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

Metric-based, goal-oriented mesh adaptation using machine learning

This paper presents a machine-learning approach for determining the optimal anisotropy in a computational mesh, in the context of an output-based adaptive solution procedure. Artificial neural networks are used to predict the desired element aspect ratio from readily accessible features of the primal and adjoint solutions. Whereas the sizing of the element is still based on an adjoint-weighted residual error estimate, the network augments this information with element stretching magnitude and direction, at lower computational and implementation costs compared to a more rigorous approach: mesh optimization through error sampling and synthesis (MOESS). The network is trained to provide anisotropy information in the form of a normalized metric field, computed from primal, adjoint, and error indicator features. MOESS-optimized meshes for a variety of steady aerodynamic flows governed by the Reynolds-averaged Navier-Stokes equations in two dimensions provide data for training several multi-layer perceptron networks, which differ in size and inputs. The networks are then deployed and tested by driving complete mesh adaptation sequences, and the results show improvements in mesh efficiency compared to pure primal Hessian-based anisotropy detection and in many cases to MOESS itself.

Goal-oriented error estimation and mesh adaptation for shallow water modelling

This study presents a novel goal-oriented error estimate for the nonlinear shallow water equations solved using a mixed discontinuous/continuous Galerkin approach. This error estimator takes account of the discontinuities in the discrete solution and is used to drive two metric-based mesh adaptation algorithms: one which yields isotropic meshes and another which yields anisotropic meshes. An implementation of these goal-oriented mesh adaptation algorithms is described, including a method for approximating the adjoint error term which arises in the error estimate. Results are presented for simulations of two model tidal farm configurations computed using the Thetis coastal ocean model (Kärnä et al. in Geosci Model Dev 11(11):4359–4382, 2018). Convergence analysis indicates that meshes resulting from the goal-oriented adaptation strategies permit accurate QoI estimation using fewer computational resources than uniform refinement.

Finite element modelling of geophysical electromagnetic data with goal-oriented hr-adaptivity

Large discontinuities (jumps) in coefficients often appear when modelling physical problems involving inhomogeneous media. An example of this is the geophysical electromagnetic (EM) problem, where these jumps occur at interfaces which separate regions of (high) conductivity contrasts. These interfaces, along with other problem features, such as singular EM sources, motivate the use of adaptive mesh refinement to improve the efficiency of the solution of forward problems. Also, pointwise observations are made in geophysical EM surveys, which motivates the use of goal-oriented mesh adaptivity. In this work, we study the combined application of h- and r-refinements for the modelling of geophysical EM data. The proposed hr-adaptivity algorithm is thoroughly investigated in 1D, and then extended, and numerically validated, in 2D, using simple examples as well as a realistic model with irregular interfaces. In 1D, the steady-state diffusion and Helmholtz equations, which are commonly solved for the EM scalar and vector potentials, respectively [16], constitute the physical partial differential equations (PPDEs). In 2D, the Helmholtz equation is used as the PPDE. Additionally, a real 2D problem with a benchmark model is considered where the transverse electric (TE) mode of Maxwell’s equations is used as the PPDE. The r-refinements are based on an equidistribution principle and variational methods, for the 1D and 2D cases, respectively, which lead to mesh PDEs (MPDEs). The coupled PPDE and MPDE are solved in an iterative manner to enhance the accuracy of the PPDE solution by improving the equidistribution of a monitor function based on a posteriori error estimates. The results, in 1D, with two hierarchical error–based monitor functions and a residual error–based monitor function display the similarity between these functions in terms of the accuracy that could be achieved. In both 1D and 2D, comparisons are made between global and goal-oriented adaptivity which show the advantage of goal-oriented error estimates in gaining higher accuracy at a target, compared to global error estimates. In 2D, the results also demonstrate the higher efficiency of hr-adaptivity compared with pure h-refinement, in terms of computation time.

Duality based error estimation in the presence of discontinuities

Goal-oriented mesh adaptation, in particular using the dual-weighted residual (DWR) method, is known in many cases to produce very efficient meshes. For obtaining such meshes the (numerical) solution of an adjoint problem is needed to weight the residuals appropriately with respect to their relevance for the overall error. For hyperbolic problems already the weak primal problem requires in general an additional entropy condition to assert uniqueness of solutions; this difficulty is also reflected when considering adjoints to hyperbolic problems involving discontinuities where again an additional requirement (reversibility) is needed to select appropriate solutions.Within this article, an approach to the DWR method for hyperbolic problems based on an artificial viscosity approximation is proposed. It is discussed why the proposed method provides a well-posed dual problem, while a direct, formal, application of the dual problem does not. Moreover, we will discuss a further, novel, approach in which the forward problem need not be modified, thus allowing for an unchanged forward solution. The latter procedure introduces an additional residual term in the error estimation, accounting for the inconsistency between primal and dual problem.Finally, the effectivity of the extended error estimator, assessing the global error by a suitable functional of interest, is tested numerically; and the advantage over a formal estimator approach is demonstrated.

Multigoal-oriented error estimates for non-linear problems

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

Computational modeling of geological materials is challenging. Firstly, they are heterogeneous with numerous uncertainties in the input parameters and secondly, the computational cost of modeling geological structures is time consuming due to the large and different length scales involved. In this article, we propose an efficient computational method for heterogeneous geological materials based on goal oriented error estimation and adaptive mesh refinement. Instead of estimating the error in a specific norm, the proposed novel error estimation approach which is called dual-weighted residual error estimation, approximates the error with respect to the quantity of interest. The dual-weighted residual error estimation is a dual-based scheme which requires an adjoint problem. The adjoint or dual problem is described by defining the quantity of interest in a functional form. Then by solving the primal and dual problems, errors in terms of the specified quantities are calculated. In many applications in engineering geology, the material is heterogeneous. In such cases, the material properties can be regarded as random fields. This variety of material properties leads to non-uniform distributions of the solution gradients, e.g. stresses. Therefore, it is vital to apply a reliable error estimation approach to be able to do efficiently the mesh-adaptivity procedure with regard to varying material parameters with pre-defined correlation lengths. Hence, the proposed error estimator is extended by accounting for a random field model to describe the material properties. Local estimated errors are exploited in order to accomplish the mesh adaptivity procedure. The goal-oriented mesh adaptivity controls the local errors in terms of the prescribed quantities. Both refinement and coarsening processes are applied to raise the efficiency. The performance of the proposed computational approach is demonstrated for several examples.

Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity

In this study, a posteriori error estimation and goal-oriented mesh adaptivity are developed for phase-field fracture propagation. Goal functionals are computed with the dual-weighted residual (DWR) method, which is realized by a recently introduced novel localization technique based on a partition-of-unity (PU). This technique is straightforward to apply since the weak residual is used. The influence of neighboring cells is gathered by the PU. Consequently, neither strong residuals nor jumps over element edges are required. Therefore, this approach facilitates the application of the DWR method to coupled (nonlinear) multiphysics problems such as fracture propagation. These developments then allow for a systematic investigation of the discretization error for certain quantities of interest. Specifically, our focus on the relationship between the phase-field regularization and the spatial discretization parameter in terms of goal functional evaluations is novel.

Finite-volume goal-oriented mesh adaptation for aerodynamics using functional derivative with respect to nodal coordinates

A new goal-oriented mesh adaptation method for finite volume/finite difference schemes is extended from the structured mesh framework to a more suitable setting for adaptation of unstructured meshes. The method is based on the total derivative of the goal with respect to volume mesh nodes that is computable after the solution of the goal discrete adjoint equation. The asymptotic behaviour of this derivative is assessed on regularly refined unstructured meshes. A local refinement criterion is derived from the requirement of limiting the first order change in the goal that an admissible node displacement may cause. Mesh adaptations are then carried out for classical test cases of 2D Euler flows. Efficiency and local density of the adapted meshes are presented. They are compared with those obtained with a more classical mesh adaptation method in the framework of finite volume/finite difference schemes 46. Results are very close although the present method only makes usage of the current grid.

An adjoint-truncation error based approach for goal-oriented mesh adaptation

We present the development and validation of an output-based mesh adaptation approach which combines the adjoint methodology with the τ-estimation technique for truncation error estimation. The approach relies on the use of an auxiliary coarse mesh to estimate the functional error and drive the mesh adaptation. The functional is estimated with an error representation formula given by the inner product of the adjoint solution and the discrete residual of the exact solution, which is approximated by the τ-estimation method. This method avoids the use of an embedded fine grid which is the common approach in the adjoint-based adaptation methods. Furthermore, it also allows to reduce the computational time overhead per adaptation step because the adjoint solution is obtained on a coarser mesh level. The effectiveness of the method is demonstrated in the framework of unstructured-mesh finite volume discretization for two- and three-dimensional Euler flows and comparisons against several mesh adaptation approaches are shown.

Goal-oriented adaptive composite discontinuous Galerkin methods for incompressible flows

In this article we consider the application of goal-oriented mesh adaptation to problems posed on complicated domains which may contain a huge number of local geometrical features, or micro-structures. Here, we exploit the composite variant of the discontinuous Galerkin finite element method based on exploiting finite element meshes consisting of arbitrarily shaped element domains. Adaptive mesh refinement is based on constructing finite element partitions of the domain consisting of agglomerated elements which belong to different levels of an underlying hierarchical tree data structure. As an example of the application of these techniques, we consider the numerical approximation of the incompressible Navier–Stokes equations. Numerical experiments highlighting the practical performance of the proposed refinement strategy will be presented.

Adaptive stabilized finite element framework for simulation of vocal fold turbulent fluid-structure interaction and towards aeroacoustics

As a step toward building a more complete model of voice production mechanics, we assess the feasibility of a fluid-structure simulation of the vocal fold mechanics in the Unicorn incompressible Unified Continuum framework. The Unicorn framework consists of conservation equations for mass and momentum, a phase function selecting solid or fluid constitutive laws, a convection equation for the phase function and moving mesh methods for tracking the interface, and discretization through an adaptive stabilized finite element method. The framework has been validated for turbulent flow for both low and high Reynolds numbers and has the following features: implicit turbulence modeling (turbulent dissipation only occurs through numerical stabilization), goal-oriented mesh adaptivity, strong, implicit fluid-structure coupling, and good scaling on massively parallel computers. We have applied the framework for turbulent fluid-structure interaction simulation of vocal folds, and present recent results. Initial steps toward aeroacoustics have been carried out in the framework (exhaust system and landing gear applications), and will be presented as well as the current state of aeroacoustic modeling for the human voice in the framework.

Goal oriented mesh adaptation using total derivative of aerodynamic functions with respect to mesh coordinates – With applications to Euler flows

In aeronautical CFD, engineers require accurate predictions of the forces and moments but they are less concerned with the accuracy of the detailed flow-field. Hence, the so-called “goal oriented” mesh adaptation strategies have been introduced to get satisfactory values of functional outputs at an acceptable cost, using local node displacement and insertion of new points rather than mesh refinement guided by uniform accuracy. Most often, such methods involve the adjoint vector of the function of interest. Our purpose is precisely to present new goal oriented mesh adaptation strategies in the framework of finite-volume schemes and a discrete adjoint method. It is based on the total derivative of the goal with respect to (w.r.t.) mesh nodes. More precisely, a projection of the goal derivative, removing all components corresponding to geometrical changes in the solid walls or the support of the output, is used to adapt the meshes either by inserting new nodes or by displacing current mesh nodes. The methods are assessed in the case of 2D and 3D Euler flow computations.

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space–time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.