The estimation of truncation error by tau-estimation revisited

The aim of this paper was to accurately estimate the local truncation error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local truncation error using the @t-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the truncation error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration error

Mesh refinement in finite element analysis by minimization of the stiffness matrix trace

Most finite element packages provide means to generate meshes automatically. However, the user is usually confronted with the problem of not knowing whether the mesh generated is appropriate for the problem at hand. Since the accuracy of the finite element results is mesh dependent, mesh selection forms a very important step in the analysis. Indeed, in accurate analyses, meshes need to be refined or rezoned until the solution converges to a value so that the error is below a predetermined tolerance. A-posteriori methods use error indicators, developed by using the theory of interpolation and approximation theory, for mesh refinements. Some use other criterions, such as strain energy density variation and stress contours for example, to obtain near optimal meshes. Although these methods are adaptive, they are expensive. Alternatively, a priori methods, until now available, use geometrical parameters, for example, element aspect ratio. Therefore, they are not adaptive by nature. An adaptive a-priori method is developed. The criterion is that the minimization of the trace of the stiffness matrix with respect to the nodal coordinates, leads to a minimization of the potential energy, and as a consequence provide a good starting mesh. In a few examples the method is shown to provide the optimal mesh. The method is also shown to be relatively simple and amenable to development of computer algorithms. When the procedure is used in conjunction with a-posteriori methods of grid refinement, it is shown that fewer refinement iterations and fewer degrees of freedom are required for convergence as opposed to when the procedure is not used. The mesh obtained is shown to have uniform distribution of stiffness among the nodes and elements which, as a consequence, leads to uniform error distribution. Thus the mesh obtained meets the optimality criterion of uniform error distribution

Numerical dissipation vs. subgrid-scale modelling for large eddy simulation

This study presents an alternative way to perform large eddy simulation based on a targeted numerical dissipation introduced by the discretization of the viscous term. It is shown that this regularisation technique is equivalent to the use of spectral vanishing viscosity. The flexibility of the method ensures high-order accuracy while controlling the level and spectral features of this purely numerical viscosity. A Pao-like spectral closure based on physical arguments is used to scale this numerical viscosity a priori. It is shown that this way of approaching large eddy simulation is more efficient and accurate than the use of the very popular Smagorinsky model in standard as well as in dynamic version. The main strength of being able to correctly calibrate numerical dissipation is the possibility to regularise the solution at the mesh scale. Thanks to this property, it is shown that the solution can be seen as numerically converged. Conversely, the two versions of the Smagorinsky model are found unable to ensure regularisation while showing a strong sensitivity to numerical errors. The originality of the present approach is that it can be viewed as implicit large eddy simulation, in the sense that the numerical error is the source of artificial dissipation, but also as explicit subgrid-scale modelling, because of the equivalence with spectral viscosity prescribed on a physical basis

Approximation of full-boundary data from partial-boundary electrode measurements

Measurements on a subset of the boundary are common in electrical impedance tomography, especially any electrode model can be interpreted as a partial boundary problem. The information obtained is different to full-boundary measurements as modeled by the ideal continuum model. In this study we discuss an approach to approximate full-boundary data from partial-boundary measurements that is based on the knowledge of the involved projections. The approximate full-boundary data can then be obtained as the solution of a suitable optimization problem on the coefficients of the Neumann-to-Dirichlet map. By this procedure we are able to improve the reconstruction quality of continuum model based algorithms, in particular we present the effectiveness with a D-bar method. Reconstructions are presented for noisy simulated and real measurement data

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Improving the quality of finite volume meshes through genetic optimisation

Author's accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00366-015-0423-0Mesh quality issues can have a substantial impact on the solution process in Computational Fluid Dynamics (CFD), leading to poor quality solutions, hindering convergence and in some cases, causing the solution to diverge. In many areas of application, there is an interest in automated generation of finite volume meshes, where a meshing algorithm controlled by pre- specified parameters is applied to a pre-existing CAD geometry. In such cases the user is typically confronted with a large number of controllable parameters, and ad- justing these takes time and perserverence. The process can however be regarded as a multi-input and possi- bly multi-objective optimisation process which can be optimised by application of Genetic Algorithm tech- niques. We have developed a GA optimisation code in the language Python, including an implementation of the NGSA-II multi-objective optimisation algorithm, and applied to control the mesh generation process us- ing the snappyHexMesh automated mesher in Open- FOAM. We demonstrate the results on three selected cases, demonstrating significant improvement in mesh quality in all cases

Target Element Sizes For Finite Element Tidal Models From A Domain-wide, Localized Truncation Error Analysis Incorporating Botto

A new methodology for the determination of target element sizes for the construction of finite element meshes applicable to the simulation of tidal flow in coastal and oceanic domains is developed and tested. The methodology is consistent with the discrete physics of tidal flow, and includes the effects of bottom stress. The method enables the estimation of the localized truncation error of the nonconservative momentum equations throughout a triangulated data set of water surface elevation and flow velocity. The method\u27s domain-wide applicability is due in part to the formulation of a new localized truncation error estimator in terms of complex derivatives. More conventional criteria that are often used to determine target element sizes are limited to certain bathymetric conditions. The methodology developed herein is applicable over a broad range of bathymetric conditions, and can be implemented efficiently. Since the methodology permits the determination of target element size at points up to and including the coastal boundary, it is amenable to coastal domain applications including estuaries, embayments, and riverine systems. These applications require consideration of spatially varying bottom stress and advective terms, addressed herein. The new method, called LTEA-CD (localized truncation error analysis with complex derivatives), is applied to model solutions over the Western North Atlantic Tidal model domain (the bodies of water lying west of the 60° W meridian). The convergence properties of LTEACD are also analyzed. It is found that LTEA-CD may be used to build a series of meshes that produce converging solutions of the shallow water equations. An enhanced version of the new methodology, LTEA+CD (which accounts for locally variable bottom stress and Coriolis terms) is used to generate a mesh of the WNAT model domain having 25% fewer nodes and elements than an existing mesh upon which it is based; performance of the two meshes, in an average sense, is indistinguishable when considering elevation tidal signals. Finally, LTEA+CD is applied to the development of a mesh for the Loxahatchee River estuary; it is found that application of LTEA+CD provides a target element size distribution that, when implemented, outperforms a high-resolution semi-uniform mesh as well as a manually constructed, existing, documented mesh