Modal‐based goal‐oriented error assessment for timeline‐dependent quantities in transient dynamics

This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline‐dependent quantities) is proposed. These quantities are scalar time‐dependent outputs of the transient solution, which are better suited to time‐dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline‐dependent quantities of interest. The key ingredient is the modal‐based approximation of the associated adjoint problems, which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post‐processed to produce an enhanced solution, requiring only one spatial post‐process for each vibration mode and using the time‐harmonic hypothesis to recover the time dependence. Thus, the proposed goal‐oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low. Copyright © 2013 John Wiley & Sons, Ltd.     

Output-based mesh optimization for hybridized and embedded discontinuous Galerkin methods

This paper presents a method for optimizing computational meshes for the prediction of scalar outputs when using hybridized and embedded discontinuous Galerkin (HDG/EDG) discretizations. Hybridization offers memory and computational time advantages compared to the standard discontinuous Galerkin (DG) method through a decoupling of elemental degrees of freedom and the introduction of face degrees of freedom that become the only globally coupled unknowns. However, the additional equations of weak flux continuity on each interior face introduce new residuals that augment output error estimates and complicate existing element-centric mesh optimization methods. This work presents techniques for converting face-based error estimates to elements and sampling their reduction with refinement in order to determine element-specific anisotropic convergence rate tensors. The error sampling uses fine-space adjoint projections and does not require additional solves on subelements. Together with a degree-of-freedom cost model, the error models drive metric-based unstructured mesh optimization. Adaptive results for inviscid and viscous two-dimensional flow problems demonstrate (i) improvement of EDG mesh optimality when using error models that incorporate face errors, (ii) the relative insensitivity of HDG mesh optimality to the incorporation of face errors, and (iii) degree of freedom and computational-time benefits of hybridized methods, particularly EDG, relative to DG.     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

We introduce a port (interface) approximation and a posteriori error bound framework for a general component‐based static condensation method in the context of parameter‐dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non‐conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization. Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds. Copyright © 2013 John Wiley & Sons, Ltd.     

An evaluation of the computational error near the boundary withmagnetostatic field calculation by BEM

A 3-D magnetostatic field is calculated using the reduced scalar potential method for a two-region model: a current-free iron region with linear and isotropic property, and an air region including the source domain. An unstable computational error near the boundary is investigated from the viewpoints of numerical integration and discretization. Specifically it is shown that near the boundary elements, the calculated results of the magnetic flux density often contain an unstable error. The error is affected by the fineness of the discretization, the point number for Gaussian quadrature, and the distance from the boundary elements. It is found that close integration of the internal field calculation effectively removes the unstable computational error     

A new numerical method for the boundary optimal control problems of the heat conduction equation

A new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate‐gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct differentiation method or the adjoint variable method. The direct differentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very efficient, the adjoint variable does not have sufficient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this difficulty with the adjoint variable method by introducing a partial differential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems efficiently with sufficient accuracy. Copyright © 2001 John Wiley & Sons, Ltd.     

Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero‐velocities in solid material. The resulting parameter optimization problem is solved by a non‐linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design‐dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non‐physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady‐state counterparts. Copyright © 2011 John Wiley & Sons, Ltd.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

Cost reduction techniques for the design of non‐linear flapping wing structures

This work details a computational framework for gradient‐based optimization of a non‐linear flapping wing structure with a large number of design variables, where analytical sensitivities of the unsteady finite element system are computed using the adjoint method. Two techniques are used to reduce the large computational cost of this structural design process. The first projects the finite element system onto a reduced basis of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time‐periodic response. Results are given in terms of the trade‐off between accuracy and computational efficiency of these methods for both system response and adjoint computations, for a variety of mesh/time step refinements, degrees of non‐linearity (i.e. weakly or strongly non‐linear), and harmonic content. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization. Published in 2011 by John Wiley & Sons, Ltd.     

Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

The parametric analysis of electric grids requires carrying out a large number of power flow computations. The different parameters describe loading conditions and grid properties. In this framework, the proper generalized decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to (1) iterating algebraic solver, (2) number of terms in the separable greedy expansion, and (3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal‐oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end‐user. The paper discusses how to compute the goal‐oriented error estimates. This requires linearizing the error equation and the quantity of interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

This paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparison of the continuous and discrete adjoint approach extended based on the standard lattice Boltzmann method in flow field inverse optimization problems

The objective of this research is the numerical implementation and comparison between the performance of the continuous and discrete adjoint Lattice Boltzmann (LB) methods in optimization problems of unsteady flow fields. For this purpose, a periodic two-dimensional incompressible channel flow affected by the constant and uniform body forces is considered as the base flow field. The standard LB method and D2Q9 model are employed to solve the flow field. Moreover, the inverse optimization of the selected flow field is defined by considering the body forces as the design variables and the sum of squared errors of flow field variables on the whole field as the cost function. In this regard, the continuous and discrete adjoint approaches extended based on the LB method are used to achieve the gradients of the cost function with respect to the design variables. Finally, the numerical results obtained from the continuous adjoint LB method are compared with the discrete one, and the accuracy and efficiency of them are discussed. In addition, the validity of the obtained cost function gradients is investigated by comparing with the results of the standard forward finite difference and complex step methods. The numerical results show that regardless of the implementation cost of the two approaches, the computational cost to evaluate the gradients in each optimization cycle for the discrete adjoint LB approach is slightly more than the other one but has a little higher convergence rate and needs a smaller number of cycles to converge. Besides, the gradients obtained from the discrete version have a better agreement with those of the complex step method. Eventually, based on the structural similarities of the continuous LB equation and its corresponding adjoint one and using the simple periodic and complete bounce-back boundary conditions for the LB equation, the improved boundary conditions for the continuous adjoint LB equation are presented. The numerical results show that the use of these boundary conditions instead of the original adjoint boundary conditions significantly improves the relative accuracy and also the convergence rate of the continuous adjoint LB method.     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

On the use of the adjoint operator for source reconstruction in particle transport problems

In this work, the adjoint to the transport operator is used to estimate the spatial distribution of an isotropic neutral particles source in a homogeneous one-dimensional medium, from readings of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. Simulations are performed for reconstructing Gaussian and piecewise localized sources. Numerical results indicate that the source estimates may be considered satisfactory although scalar fluxes are reconstructed more precisely.     

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

An s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order finite volume schemes for the advection–diffusion equation

A high‐order finite volume method based on piecewise interpolant polynomials is proposed to discretize spatially the one‐dimensional and two‐dimensional advection–diffusion equation. Evolution equations for the mean values of each control volume are integrated in time by a classical fourth‐order Runge–Kutta. Since our work focuses on the behaviour of the spatial discretization, the time step is chosen small enough to neglect the time integration error. Two‐dimensional interpolants are built by means of one‐dimensional interpolants. It is shown that when the degree of the one‐dimensional interpolant q is odd, the proper selection of a fixed stencil gives rise to centred schemes of order q+1. In order not to lose precision due to the change of stencil near boundaries, the degree of the interpolants close to boundaries is raised to q+1. Four test cases with small values of diffusion are integrated with high‐order methods. It is shown that the spatial discretization of the advection–diffusion equation with periodic boundary conditions leads to normal discretization matrices, and asymptotic stability must be assured to bound the spatial discretization error. Once the asymptotic stability is assured by means of the spectra of the discretization matrix, the spatial error is of the order of the truncation error. However, it is shown that the discretization of the advection–diffusion equation with arbitrary boundary conditions gives rise to non‐normal matrices. If asymptotic stability is assured, the spatial order of steady solutions is of the order of the truncation error. But, for transient processes, the order of the spatial error is determined by both the truncation error and the norm of the exponential matrix of the spatial discretization. The use of the pseudospectra of the discretization matrix is proposed as a valuable tool to analyse the transient error of the numerical solution. Copyright © 2001 John Wiley & Sons, Ltd.     

Enhanced goal‐oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

This work focuses on providing accurate low‐cost approximations of stochastic finite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte‐Carlo method for multi‐dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal‐oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments. Copyright © 2016 John Wiley & Sons, Ltd.     

Sensitivity analysis and design optimization of three‐dimensional non‐linear aeroelastic systems by the adjoint method

We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.