Masayuki Yano

An optimization-based framework for anisotropic simplex mesh adaptation

We present a general framework for anisotropic h-adaptation of simplex meshes. Given a discretization and any element-wise, localizable error estimate, our adaptive method iterates toward a mesh that minimizes error for a given degrees of freedom. Utilizing mesh-metric duality, we consider a continuous optimization problem of the Riemannian metric tensor field that provides an anisotropic description of element sizes. First, our method performs a series of local solves to survey the behavior of the local error function. This information is then synthesized using an affine-invariant tensor manipulation framework to reconstruct an approximate gradient of the error function with respect to the metric tensor field. Finally, we perform gradient descent in the metric space to drive the mesh toward optimality. The method is first demonstrated to produce optimal anisotropic meshes minimizing the L^2 projection error for a pair of canonical problems containing a singularity and a singular perturbation. The effectiveness of the framework is then demonstrated in the context of output-based adaptation for the advection-diffusion equation using a high-order discontinuous Galerkin discretization and the dual-weighted residual (DWR) error estimate. The method presented provides a unified framework for optimizing both the element size and anisotropy distribution using an a posteriori error estimate and enables efficient adaptation of anisotropic simplex meshes for high-order discretizations.

A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations

We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov--Galerkin space-time trial- and test-space norms that yield optimal and asymptotically mesh independent stability constants. We then employ an

The Importance of Mesh Adaptation for Higher-Order Discretizations of Aerodynamic Flows

hp

Plasma ionization by annularly bounded helicon waves

Petrov--Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi--Rappaz--Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based, respectively, on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and ...

An Optimization Framework for Anisotropic Simplex Mesh Adaptation: Application to Aerodynamic Flows

This work presents an adaptive framework for a higher-order discretization of the Reynolds-averaged Navier-Stokes (RANS) equations. The adaptation strategy is based on an output-based error estimate and explicit control of the degrees of freedom. Adaptation iterates toward the generation of simplex meshes that equidistribute local errors throughout the domain and provide anisotropic resolution in arbitrary orientations. Numerical experiments reveal that uniform re nement limits the performance of higher-order methods when applied to aerodynamic ows with low regularity. However, when combined with anisotropic re nement of singular features, higher-order methods can signi cantly improve computational a ordability of RANS simulations in the engineering environment. The bene t of the higher spatial accuracy is exhibited for a wide range of applications including subsonic, transonic, and supersonic ows. The higher-order simplex meshes are generated using the elasticity and the cut-cell techniques, and the competitiveness of the cut-cell method is demonstrated in terms of accuracy per degree of freedom.

A Comparison of Higher-Order Methods on a Set of Canonical Aerodynamics Applications

The general solution to the electrostatic and magnetic fields is derived with respect to the boundary conditions of a coaxial helicon plasma source. The electric field contours suggest that a simple antenna design can ionize the gas in a coaxial configuration. In addition, the power deposition as a function of excitation frequency is derived. The solution is validated by comparison with the standard cylindrical helicon plasma source. Further, a parametric study of source length, channel radius, channel width, and antenna excitation frequency are presented. This study suggests that it is possible to create a helicon plasma source with a coaxial configuration.

A space–time variational approach to hydrodynamic stability theory

Singapore-MIT Alliance for Research and Technology (Fellowship in Computational Engineering)

Generalized theory of annularly bounded helicon waves

Higher-order discretizations have the potential to reduce the computational cost required to achieve a desired error level. In this study, we consider higher-order discretizations of the conservation equations suitable for unstructured, triangular grids. In particular, the methods studied include continuous (SUPG/GLS) and classical discontinuous Galerkin (DG) finite element methods, the correction procedure via reconstruction (CPR) formulations of the DG and spectral volume methods, and cell and vertex-centered finite volume (FV) algorithms. This paper presents subsonic and supersonic, inviscid results for a canonical set of aerodynamic applications. Error convergence and computational performance of these discretizations are compared, and preliminary results indicate that the methods perform relatively similarly. When singularities are present in the flow solutions and uniformly refined meshes are used, all methods fail to achieve optimal convergence rates, and the performance benefits of the higher-order discretizations are reduced; adaptive meshing improves the efficiency of the higher-order method and recovers optimal convergence rates.

Design and Operation of an Annular Helicon Plasma Source

We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

A Reduced Basis Method for Coercive Equations with an Exact Solution Certificate and Spatio-Parameter Adaptivity: Energy-Norm and Output Error Bounds

The generalized dispersion relationship for annularly bounded helicon plasma is derived. The theory considers the effect of finite electron mass, which results in the plasma field described by superposition of classical helicon solution and the Trivelpiece-Gould (TG) wave. The solution is obtained for an insulating boundary. The solution shows that the wave structure is heavily affected by the presence of the TG wave, even when a strong axial magnetic field is applied. However, the electric field profile shows that a pair of external rf antennas can still be coupled to the strong radial electric field in the plasma to excite the wave. Moreover, the general solution permits an almost continuous spectrum of parallel wave numbers. The study of k‐f and n0‐B0 diagrams shows that the linear relation obtained using classical theory holds everywhere except at low applied magnetic field strength.