Journal articles by Artur Palha
Journal of Computational Physics, 2020
A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is pre... more A model of the three-dimensional rotating compressible Euler equations on the cubed sphere is presented. The model uses a mixed mimetic spectral element discretization which allows for the exact exchanges of kinetic, internal and potential energy via the compatibility properties of the chosen function spaces. A Strang carryover dimensional splitting procedure is used, with the horizontal dynamics solved explicitly and the vertical dynamics solved implicitly so as to avoid the CFL restriction of the vertical sound waves. The function spaces used to represent the horizontal dynamics are discontinuous across vertical element boundaries, such that each horizontal layer is solved independently so as to avoid the need to invert a global 3D mass matrix, while the function spaces used to represent the vertical dynamics are similarly discontinuous across horizontal element boundaries, allowing for the serial solution of the vertical dynamics independently for each horizontal element. The model is validated against standard test cases for baroclinic instability within an otherwise hydrostatically and geostrophically balanced atmosphere, and a non-hydrostatic gravity wave as driven by a temperature perturbation.
Proceedings of The Institution of Civil Engineers-maritime Engineering, 2010
Coastal Engineering 2008 - Proceedings of the 31st International Conference, 2009
Book chapters by Artur Palha
Lecture Notes in Computational Science and Engineering, 2013
ABSTRACT The relation between physics, its description in terms of partial differential equations... more ABSTRACT The relation between physics, its description in terms of partial differential equations and geometry is explored in this paper. Geometry determines the correct weak formulation in finite element methods and also dictates which basis functions should be employed to obtain discrete well-posedness.
This paper describes a mimetic spectral element method on curvilinear grids applied to the Poisso... more This paper describes a mimetic spectral element method on curvilinear grids applied to the Poisson equation. The Poisson equation is formulated in terms of differential forms. The spectral basis functions in which the differential forms are expressed lead to a metric free discrete representation of the gradient and the divergence operator. Using the fact that the pullback operator commutes with the wedge product and the exterior derivative leads to a mimetic spectral element formulation on curvilinear grids which displays exponential convergence and satisfies the divergence exactly. The robustness of the proposed scheme will be demonstrated for a sample problem for which exponential convergence is obtained.
This paper describes a mimetic spectral element formulation for the Poisson equation on quadrilat... more This paper describes a mimetic spectral element formulation for the Poisson equation on quadrilateral elements. Two dual grids are employed to represent the two first order equations. The discrete Hodge operator, which connects variables on these two grids, is the derived Hodge operator obtained from the wedge product and the inner-product. The gradient operator is not discretized directly, but derived from the symmetry relation between gradient and divergence on dual meshes, as proposed by Hyman et al., [5], thus ensuring a symmetric discrete Laplace operator. The resulting scheme is a staggered spectral element scheme, similar to the staggering proposed by Kopriva and Kolias, [6]. Different integration schemes are used for the various terms involved. This scheme is equivalent to a least-squares formulation which minimizes the difference between the dual velocity representations. This generates the discrete Hodge-⋆ operator. The discretization error of this schemes equals the interpolation error.
Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical sc... more Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimize the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme, see, for example, Cai et al. (J. Numer. Anal. 34:425–454, 1997).
Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical sc... more Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimise the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme.
Preprints by Artur Palha
Currently, Eulerian flow solvers are very efficient in accurately resolving flow structures near ... more Currently, Eulerian flow solvers are very efficient in accurately resolving flow structures near solid boundaries. On the other hand, they tend to be diffusive and to dampen high-intensity vortical structures after a short distance away from solid boundaries. The use of high order methods and fine grids, although alleviating this problem, gives rise to large systems of equations that are expensive to solve. Lagrangian solvers, as the regularized vortex particle method, have shown to eliminate (in practice) the diffusion in the wake. As a drawback, the modelling of solid boundaries is less accurate, more complex and costly than with Eulerian solvers (due to the isotropy of its computational elements). Given the drawbacks and advantages of both Eulerian and Lagrangian solvers the combination of both methods, giving rise to a hybrid solver, is advantageous. The main idea behind the hybrid solver presented is the following. In a region close to solid boundaries the flow is solved with a...
Conference proceedings by Artur Palha
This paper describes the experiments performed at the National Laboratory for Civil Engineering (... more This paper describes the experiments performed at the National Laboratory for Civil Engineering (LNEC) aiming at simulating, in a flume, the wave propagation along a constant slope bottom that ends on a sea wall coastal defence structure, a common structure employed in the ...
Papers by Artur Palha
We present a discretization of the linear advection of differential forms on bounded domains. The... more We present a discretization of the linear advection of differential forms on bounded domains. The framework established in [4] is extended to incorporate the Lie derivative, L, by means of Cartan’s homotopy formula. The method is based on a physics-compatible discretization with spectral accuracy. It will be shown that the derived scheme has spectral convergence with local mass conservation. Artificial dispersion depends on the order of time integration. 1
In this paper we will consider two curl-curl equation in two dimensions. One curl-curl problem fo... more In this paper we will consider two curl-curl equation in two dimensions. One curl-curl problem for a scalar quantity F and one problem for a vector field E. For Dirichlet boundary conditions n×E = Ê_ on E and Neumann boundary conditions n×curl F=Ê_, we expect the solutions to satisfy E=curl F. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.
Uploads
Journal articles by Artur Palha
Book chapters by Artur Palha
Preprints by Artur Palha
Conference proceedings by Artur Palha
Papers by Artur Palha