High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes

We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as 4.7.Comment: 31 pages, 13 figure

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Numerical simulation of two and three-dimensional viscous free surface flows in partially-filled containers using a surface capturing approach

A new surface capturing method is developed for numerically simulating viscous free surface flows in partially-filled containers. The method is based on the idea that the flow of two immiscible fluids (specifically, a liquid and a gas) within a closed container is governed by the equations of motion for a laminar, incompressible, viscous, nonhomogeneous (variable density) fluid. By computing the flowfields in both the liquid and gas regions in a consistent manner, the free surface can be captured as a discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures;The numerical algorithm is developed using a conservative, implicit, finite volume discretization of the equations of motion. The algorithm incorporates the artificial compressibility method in conjunction with a dual time stepping strategy to maintain a divergence-free velocity field. A slope-limited, high order MUSCL scheme is adopted for approximating the inviscid flux terms, while the viscous fluxes are centrally differenced. Two different methods are considered for solving the resulting block-banded system of equations;The capabilities of the surface capturing method are demonstrated by calculating solutions to several challenging two and three-dimensional problems. The first test case, the two-dimensional broken dam problem, is considered in detail. Results are presented for several grid sizes, upwind schemes, and limiters, and are compared to experimental data from the literature. The solutions employing high order upwind interpolants and a compressive minmod limiter on the density are found to yield the best results. The two-dimensional, viscous Rayleigh-Taylor instability is examined next. Solutions for a density ratio of two are computed for various Reynolds numbers. Computed perturbation growth rates are shown to be in good agreement with theoretical predictions. Results for the three-dimensional broken dam problem are then presented. The computed free surface motions are found to be quite similar to the two-dimensional case. Finally, two cases involving axisymmetric spin-up in a spherical container are studied. The computed free surface shapes are found to exhibit the characteristic parabolic profiles as steady state conditions are approached

Analysis and optimization of film cooling effectiveness

In the first part, an optimization strategy is described that combines high-fidelity simu- lations with response surface construction, and is applied to pulsed film cooling for turbine blades. The response surface is constructed for the film cooling effectiveness as a function of duty cycle, in the range of DC between 0.05 and 1, and pulsation frequency St in the range of 0.2-2, using a pseudo-spectral projection method. The jet is fully modulated and the blowing ratio, when the jet is on, is 1.5 in all cases. Overall 73 direct numerical sim- ulations (DNS) using spectral element method were performed to sample the film cooling effectiveness on a Clenshaw-Curtis grid in the design space. It is observed that in the parameter space explored a global optimum exists, and in the present study, the best film cooling effectiveness is found at DC = 0.14 and St = 1.03. In the same range of DC and St, four other local optimums were found. The gradient-based optimization algorithms are argued to be unsuitable for the current problem due to the non-convexity of the objective function. In the second part, the effect of randomness of blowing ratio on film cooling performance is investigated by combining direct numerical simulations with a stochastic collocation ap- proach. The blowing ratio variations are assumed to have a truncated Gaussian distribution with mean of 0.3 and the standard variation of approximately 0.1. The parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) with five elements where general polynomial chaos of order 3 is used in each element. Direct numerical simula- tions were carried out using spectral/hp element method to sample the governing equations in space and time. The probability density function of the film cooling effectiveness was obtained and the standard deviation of the adiabatic film cooling effectiveness on the blade surface was calculated. A maximum standard deviation of 15% was observed in the region within a four-jet-diameter distance downstream of the exit hole. The spatially-averaged adiabatic film cooling effectiveness was 0.23 0.02. The calculation of all the statistical properties were carried out as off-line post-processing. Overall the computational strategy is shown to be very effective with the total computational cost being equivalent to solving twenty independent direct numerical simulations that are performed concurrently. In the third part, an accurate and efficient finite difference method for solving the incompressible Navier-Stokes equations on curvilinear grids is developed. This method combines the favorable features of the staggered grid and semi-staggered grid approaches. A novel symmetric finite difference discretization of the Poisson-Neumann problem on curvilinear grids is also presented. The validity of the method is demonstrated on four benchmark problems. The Taylor-Green vortex problem is solved on a curvilinear grid with highly skewed cells and a second-order convergence in .-norm is observed. The mixed convection in a lid-driven cavity is solved on a highly curvilinear grid and excellent agreement with literature is obtained. The results for flow past a cylinder are compared with the existing experimental data in the literature. As the fourth case, three dimensional time-dependent incompressible flow in a curved tube is solved. The predictions agree well with the measured data, and validate the approach used

Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme