Marius Paraschivoiu

A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations

We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second-order coercive linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic ‘energy’ reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine ‘truth-mesh’ discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid-flux candidate Lagrange multipliers generated by a K-element coarser ‘working-mesh’ approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomain-local, symmetric Neumann problems. The technique is illustrated for the convection-diffusion and linear elasticity equations.

A hierarchical duality approach to bounds for the outputs of partial differential equations

Abstract We present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to (finiteelement discretizations of) symmetric or nonsymmetric coercive linear partial differential equations. The method is based upon the construction of an augmented Lagrangian which integrates (i) a quadratic ‘energy’ reformulation of the desired output as the objective to be minimized, with (ii) the finite-element equilibrium equations and (conforming) ‘hybridized’ intersubdomain continuity conditions as the constraints to be satisfied. The bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate Lagrange multipliers generated by a less expensive approximation, such as a low-dimensional finite-element discretization. As in many a posteriori error estimation techniques, the bound calculation requires only the solution of subdomain-local symmetric (Neumann) problems on the refined ‘truth’ mesh. The technique is presented and illustrated for the case of the one-dimensional convection-diffusion equation.

Second order accurate volume tracking based on remapping for triangular meshes

This paper presents a second order accurate piecewise linear volume tracking based on remapping for triangular meshes. This approach avoids the complexity of extending unsplit second order volume of fluid algorithms, advection methods, on triangular meshes. The method is based on Lagrangian-Eulerian (LE) methods; therefore, it does not deal with edge fluxes and corner fluxes, flux corrections, as is typical in advection algorithms. The method is constructed of three parts: a Lagrangian phase, a reconstruction phase and a remapping phase. In the Lagrangian phase, the original, Eulerian, grid is projected along trajectories to obtain Lagrangian grids. In practice, this projection is handled through the time integration of velocity field for grid vertices at each time step. The reconstruction is based on truncating the volume material polygon for each Lagrangian mixed grid. Since in piecewise linear approximation, the interface is represented by a segment line, the polygon material truncation is mainly finding the segment interface. Finding the segment interface is calculating the line normal and line constant at each multi-fluid cell. Details of applying two normal calculation methods, differential and geometric least squares (GLS) methods, are given. While the GLS method exhibits second order accurate approximation in reproducing circular interfaces, the differential least squares (DLS) method results in a first order accurate representation of the interface. The last part of the algorithm which is remapping of the volume materials from the Lagrangian grid to the original one is performed by a series of polygon intersection procedures. The behavior of the algorithm is investigated for flow fields with constant interface topology and flow fields inducing large interfacial stretching and tearing. Second order accuracy is obtained if the velocity integration as well as the reconstruction steps are at least second order accurate.

Investigation of a two-dimensional spectral element method for Helmholtz's equation

A spectral element method is developed for solving the two-dimensional Helmholtzs equation, which is the equation governing time-harmonic acoustic waves. Computational cost for solving Helmholtzs equation with the Galerkin finite element method increases as the wave number increases, due to the pollution effect. Therefore a more efficient numerical method is sought. The comparison between a spectral element method and a second-order finite element method shows that the spectral element method leads to fewer grid points per wavelength and less computational cost, for the same accuracy. It also offers the same advantage as the finite element method to address complex geometry and general material property. Some simple examples are addressed and compared with the exact solutions to confirm the accuracy of the method. For unbounded problems, the symmetric perfectly matched layer (PML) method is applied to treat the non-reflecting boundary conditions. In the PML method, a fictitious absorbing layer is introduced outside the truncated boundary.

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

A finite element technique is presented for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions. The finite element discretization is effected by Crouzeix–Raviart elements, the discontinuous pressure approximation of which is central to this approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic ‘energy’ reformulation of the desired output, the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the inter-sub-domain continuity conditions on velocity. Appealing to the dual max–min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization. The Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires the solution of only local sub-domain Stokes problems on the fine mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flow rate past and the lift force on a body immersed in a channel. Copyright

Low Reynolds Number Vertical Axis Wind Turbine for Mars

A low Reynolds number wind turbine is designed to extract the power from wind energy on Mars. As compared to solar cells, wind turbine systems have an advantage on Mars, as they can continuously produce power during dust storms and at night. The present work specifically addresses the design of a 500 W Darrieus-type straight-bladed vertical-axis wind turbine (S-VAWT) considering the atmospheric conditions on Mars. The thin atmosphere and wind speed on Mars result in low Reynolds numbers (2000–80000) representing either laminar or transitional flow over airfoils, and influences the aerodynamic loads and performance of the airfoils. Therefore a transitional model is used to predict the lift and drag coefficients for transitional flows over airfoils. The transitional models used in the present work combine existing methods for predicting the onset and extent of transition, which are compatible with the Spalart-Allmaras turbulence model. The model is first validated with the experimental predictions reported in the literature for an NACA 0018 airfoil. The wind turbine is designed and optimized by iteratively stepping through the following tasks: rotor height, rotor diameter, chord length, and aerodynamic loads. The CARDAAV code, based on the “Double-Multiple Streamtube” model, is used to determine the performances and optimize the various parameters of the straight-bladed vertical-axis wind turbine.

Detached-Eddy Simulation of a Wing Tip Vortex at Dynamic Stall Conditions

The behavior of the tip vortex behind a square NACA0015 wing was numerically investigated. The problems studied include the stationary and the oscillating wings at static and dynamic stall conditions. Reynolds-averaged Navier-Stokes and detached-eddy simulation schemes were implemented. Vortex structures predicted by Reynolds-averaged Navier-Stokes were mainly diffused while detached-eddy simulation was able to produce qualitatively and quantitatively better results as compared to the experimental data. The breakup of the tip vortex, which started at the end of the upstroke and continued to the middle of the downstroke over an oscillation cycle, was observed in detached-eddy simulation data.

Bayesian-validated computer-simulation surrogates for optimization and design: error estimates and applications

We present a Bayesian-validated surrogate framework which permits economical and reliable integration of large-scale simulations into engineering design and optimization. In the surrogate approach, the large-scale simulation is evoked only to construct and validate a simplified input-output model; this simplified input-output model then serves as a simulation surrogate in subsequent engineering optimization studies. The distinguishing features of our approach are: sequential statistical sampling procedures which permit both efficient adaptive surrogate construction and rigorous ‘probably-approximately-correct’ surrogate validation; and validation-based non-parametric a posteriori error estimates which precisely quantify the effect of surrogate-for-simulation substitution on system predictability and optimality. In this paper we discuss recent improvements and extensions to our construction-validation algorithms and a posteriori error framework, and present several illustrative applications in heat transfer and fluid mechanics.

Fast Bounds for Outputs of Partial Differential Equations

We present a framework for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second—order coercive partial differential equations. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic “energy” reformulation of the desired output, and the constraint is a generalized representation of the finite element equilibrium equations; the bounds are then derived by evoking the dual max-min problem for appropriately chosen candidate Lagrange multipliers. Computational efficiency is ensured by proper selection of the energy functional and generalized representation. A wide variety of methods can be developed from the general formulation; we consider here two particular techniques, a hybridization approach, and a reduced—basis approach. Examples include convection—diffusion, linear elasticity, the incompressible Stokes equations, and a nonlinear conduction problem.

Self-Adaptive Upwinding for Large Eddy Simulation of Turbulent Flows on Unstructured Elements

A self-adaptive upwinding method for large eddy simulation is proposed to reduce the numerical dissipation of a low-order numerical scheme on unstructured elements. This method is used to extend an existing Reynolds-averaged Navier-Stokes code to a large eddy simulation code by adjusting the contribution of the upwinding term to the convective flux. This adjustment is essentially controlled by the intensity of the local wiggle and reduces the upwind contribution in the Roe-MUSCL scheme. First, the stability characteristic of the new scheme is studied using a channel flow stability test. It is essential to ensure that the proposed scheme is able to adjust upwinding in the presence of very high gradients and that it prohibits the divergence of the simulation. Second, the decaying isotropic turbulence is simulated to study the capability of the new scheme to generate the suitable decaying rate for the total kinetic energy and its influence over the slope of the energy spectrum at different computational times. Finally, the flow separation phenomenon over a NACA0025 profile is numerically investigated and results are compared with experimental data.