High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101?C3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.     

Numerical techniques for parallel dynamics in electromagnetic gyrokinetic Vlasov simulations

Numerical techniques for parallel dynamics in electromagnetic gyrokinetic simulations are introduced to regulate unphysical grid-size oscillations in the field-aligned coordinate. It is found that a fixed boundary condition and the nonlinear mode coupling in the field-aligned coordinate, as well as numerical errors of non-dissipative finite difference methods, produce fluctuations with high parallel wave numbers. The theoretical and numerical analyses demonstrate that an outflow boundary condition and a low-pass filter efficiently remove the numerical oscillations, providing small but acceptable errors of the entropy variables. The new method is advantageous for quantitative evaluation of the entropy balance that is required for obtaining a steady state in gyrokinetic turbulence.     

Nonlinear solution adaptive structured grids via heat-conduction analogies

This paper discusses an adaptive grid approach, developed using Fortran 77, on quadrilateral meshes for the Euler and Navier-Stokes solvers. Solution adaptation is through two nonlinear heat-conduction analogies applied directly on a two-dimensional surface using the finite volume method. Clustering of the grid generated is controlled by the conductivity in the computational domain, which is related arbitrarily to the geometrical curvature and flow gradient. Three levels of “multigrid” approach are implemented to accelerate convergence as a grid refinement process. The grid quality is accessed by a histogram analysis of maximum angle and aspect ratio distributions within the computational domain. This work assumes that interpolation errors due to numerical approximation of fluxes across the surfaces of a control volume should become significant as the skew angle and aspect ratio increases. Detailed computational results and comparisons with measured data are presented for steady transonic flow over a NACA0012 airfoil, supersonic flow through a DFVLR rotor, and a 15° ramp.     

TURBINS: An immersed boundary, Navier–Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies

An accurate, three-dimensional Navier–Stokes based immersed boundary code called TURBINS has been developed, validated and tested, for the purpose of simulating density-driven gravity and turbidity currents propagating over complex topographies. The code is second order accurate in space and third order in time, uses MPI, and employs a domain decomposition approach. It makes use of multigrid preconditioners and Krylov iterative solvers for the systems of linear equations obtained by the finite difference discretization of the governing equations. TURBINS utilizes the direct forcing variant of the immersed boundary approach and enforces the no-slip boundary condition via the first grid point inside the solid, which yields very accurate wall shear stress results. The results of test simulations are discussed for uniform flow around a circular cylinder, and for two- and three-dimensional lock-exchange gravity currents.     

An efficient parallel algorithm with application to computational fluid dynamics

When solving time-dependent partial differential equations on parallel computers using the nonoverlapping domain decomposition method, one often needs numerical boundary conditions on the boundaries between subdomains. These numerical boundary conditions can significantly affect the stability and accuracy of the final algorithm.In this paper, a stability and accuracy analysis of the existing methods for generating numerical boundary conditions will be presented, and a new approach based on explicit predictors and implicit correctors will be used to solve convection-diffusion equations on parallel computers, with application to aerospace engineering for the solution of Euler equations in computational fluid dynamics simulations. Both theoretical analyses and numerical results demonstrate significant improvement in stability and accuracy by using the new approach.     

Dimensionality Reduction on Multi-Dimensional Transfer Functions for Multi-Channel Volume Data Sets

The design of transfer functions for volume rendering is a non-trivial task. This is particularly true for multi-channel data sets, where multiple data values exist for each voxel, which requires multi-dimensional transfer functions. In this paper, we propose a new method for multi-dimensional transfer function design. Our new method provides a framework to combine multiple computational approaches and pushes the boundary of gradient-based multi-dimensional transfer functions to multiple channels, while keeping the dimensionality of transfer functions at a manageable level, i.e., a maximum of three dimensions, which can be displayed visually in a straightforward way. Our approach utilizes channel intensity, gradient, curvature and texture properties of each voxel. Applying recently developed nonlinear dimensionality reduction algorithms reduces the high-dimensional data of the domain. In this paper, we use Isomap and Locally Linear Embedding as well as a traditional algorithm, Principle Component Analysis. Our results show that these dimensionality reduction algorithms significantly improve the transfer function design process without compromising visualization accuracy. We demonstrate the effectiveness of our new dimensionality reduction algorithms with two volumetric confocal microscopy data sets.     

On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition

In this paper, a new integral vorticity boundary condition has been developed and implemented to compute solution of nonprimitive Navier–Stokes equation. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The procedure realized as explicit boundary vorticity conditions imitates the original integral equation. The main purpose of this paper is to design an algorithm which is easy to implement and versatile. This algorithm based on the new vorticity integral condition captures accurate vorticity distribution on the boundary of computational flow field and can be used for both wall bounded flows as well as flows in open domain. The approach has been arrived at without utilizing any ghost grid point outside of the computational domain. Convergence analysis of this alternative vorticity integral condition in combination with semi-discrete centered difference approximation of linear Stokes equation has been carried out. We have also computed correct pressure field near the wall, for both attached and separated boundary layer flows, by using streamfunction and vorticity field variables. The competency of the proposed boundary methodology vis-a-vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Subsequently the proposed methodology has been further validated by computing analytical solution of steady Stokes equation. Finally, it has been applied to three benchmark problems governed by the incompressible Navier–Stokes equations, viz. lid driven cavity, backward facing step and flow past a circular cylinder. The results obtained are in excellent agreement with computational and experimental results available in literature, thereby establishing efficiency and accuracy of the proposed algorithm. We were able to accurately predict both vorticity and pressure fields.     

Anatomical and dynamic volume spline model applied to facial soft tissue

Biomechanical modeling of soft tissue is a complex problem for achieving realistic surgical simulations, surgical planning, and scientific analysis. In the literature, three categories of biomechanical models: spline based models, spring models, and finite element models (FEMs) are mainly used for dealing with this problem. Among these, spline based models offer relatively fast and realistic soft tissue simulations by utilizing both the spring and FEMs. In this paper, a new dynamic volume spline model for human face skin is proposed and the performance of our model is discussed by estimating the results of facial surgery of three different patients. Face models of the patients are obtained from 3D CT/MR scans by segmenting the skull, muscle, and skin layers. In these face models, the skull and the muscle layers are considered as the rigid boundary for the skin layer and the skin layer is modeled by our dynamic volume spline. The control points of the dynamic volume spline are localized masses with viscoelastic material properties (stiffness, damping, and mass). These parameters are computed from the skin material properties that were published in the literature. Once the face models are generated, facial surgery plannings are simulated. Infact, the pre‐surgery face models are modified according to the surgical plans and the estimated post‐surgery face models are compared with the actual post‐surgery face models. Moreover, in order to discuss the performance of our dynamic volume spline model, the same analyses are performed on the post‐surgery estimations of a conventional tool. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element analysis on implicitly defined domains: An accurate representation based on arbitrary parametric surfaces

In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain’s volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm.     

Towards development of unsteady near-wall interface boundary conditions for turbulence modeling

In near-wall turbulence modeling it is required to resolve a thin layer nearby the solid boundary, which is characterized by high gradients of the solution. An accurate enough resolution of such a layer can take most computational time. The situation even becomes worse for unsteady problems. To avoid time-consuming computations, a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at the interface boundary is achieved via transfer of the boundary condition from the wall. For the first time interface boundary conditions of Robin type are derived for a model nonstationary equation which simulates the key terms of the unsteady boundary layer equations. In the case of stationary solutions the approach is automatically reduced to the technique earlier developed for the steady problems. The considered test cases demonstrate that unsteady effects can be significant for near-wall domain decomposition. In particular, they can be important in the case of the wall-function-based approach.     

Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models

We present a model order reduction approach for parametrized laminar flow problems including viscous boundary layers. The viscous effects are captured by the incompressible Navier–Stokes equations in the vicinity of the boundary layer, whereas a potential flow model is used in the outer region. By this, we provide an accurate model that avoids imposing the Kutta condition for potential flows as well as an expensive numerical solution of a global viscous model. To account for the parametrized nature of the problem, we apply the reduced basis method. The accuracy of the reduced order model is ensured by rapidly computable a posteriori error estimates. The main contributions of this paper are the combination of an offline-online splitting with the domain decomposition approach, reducing both offline and online computational loads and a new kernel interpolation method for the approximation of the stability factor in the online evaluation of the error estimate. The viability of our approach is demonstrated by numerical experiments for the section of a NACA airfoil.     

Boundary cell-based acceleration for volume ray casting

Several effective acceleration techniques for volume rendering offer efficient means to skip over empty space, providing significant speedup without affecting image quality. The effectiveness of such an approach depends on its ability to accurately estimate the object boundary inside a volume with minimal computational overhead. We propose a novel boundary cell-based acceleration technique for ray casting which skips over empty space by accurately calculating the intersection distance for each ray. Very short distance estimation time is achieved by exploiting a projection template to calculate the parallel-projection values of each boundary cell and the coherency of adjacent cells. Since no hardware acceleration is used, the projection procedure can also be efficiently parallelized. Experimental results are provided to demonstrate the performance of our new algorithm.     

Rendering and Extracting Extremal Features in 3D Fields

Visualizing and extracting three‐dimensional features is important for many computational science applications, each with their own feature definitions and data types. While some are simple to state and implement (e.g. isosurfaces), others require more complicated mathematics (e.g. multiple derivatives, curvature, eigenvectors, etc.). Correctly implementing mathematical definitions is difficult, so experimenting with new features requires substantial investments. Furthermore, traditional interpolants rarely support the necessary derivatives, and approximations can reduce numerical stability. Our new approach directly translates mathematical notation into practical visualization and feature extraction, with minimal mental and implementation overhead. Using a mathematically expressive domain‐specific language, Diderot, we compute direct volume renderings and particle‐based feature samplings for a range of mathematical features. Non‐expert users can experiment with feature definitions without any exposure to meshes, interpolants, derivative computation, etc. We demonstrate high‐quality results on notoriously difficult features, such as ridges and vortex cores, using working code simple enough to be presented in its entirety.     

An iterative method for solving 2D wave problems in infinite domains

This paper presents a new method for solving two-dimensional wave problems in infinite domains. The method yields a solution that satisfies Sommerfeld's radiation condition, as required for the correct solution of infinite domains excited only locally. It is obtained by iterations. An infinite domain is first truncated by introducing an artificial finite boundary (β), on which some boundary conditions are imposed. The finite computational domain in each iteration is subjected to actual boundary conditions and to different (Dirichlet or Neumann) fictive boundary conditions on β.     

Implementing surfactant mass balance in 2D FEM�CALE models

Practical aspects of implementing surfactant mass balance computation in finite elements models, where the model geometry shape change is captured by utilizing the arbitrary Lagrange–Eulerian method are discussed briefly. The discussion and the reported simulations are carried out in two-dimensional Cartesian coordinates. Two alternative approaches to formulating the governing equation of surfactant mass balance for solving it computationally are presented and discussed. One of the approaches is based on computing the boundary curvature and boundary tangential velocity, as well as their differentials on the boundary, directly. The other approach is based on reformulating the governing equation in order to track the proportional rate of change of local surface area. As a conclusion, it is found that though both of the presented approaches can be configured to perform adequately in terms of surfactant mass conservation, surface differentials that are necessary to compute the surface curvature and surface tangential velocity in the first one of the methods evoke numerical oscillations near those points of the boundary where it is not smooth. The text is accompanied by example simulations and figures.     

Fault detection and estimation for a class of PIDE systems based on boundary observers

In this paper, we propose a simultaneous state estimation and fault estimation approach for a class of first‐order hyperbolic partial integral differential equation systems. Specifically, we consider the multiplicative boundary actuator and sensor faults, ie, unknown fault parameters multiplying by the boundary input or boundary state (ie, output). As a consequence, two difficulties arise immediately: (1) simultaneous estimation of both plant state and faults is a nonlinear problem due to the multiplication between fault parameters and plant signals; (2) no prior information is available to determine the type (actuator or sensor) of faults. To overcome these difficulties, this paper develops adaptive fault parameter update laws and embeds the resulting laws into the plant state observer design. First, we propose new approaches to estimate actuator fault and sensor fault, respectively. Next, we develop a novel method to simultaneously estimate actuator and sensor faults. The proposed observer and update laws, designed using only one boundary measurement, ensure both state estimation and fault parameter estimation. By choosing appropriate Lyapunov functions, we prove that the estimates of state and fault parameters converge to an arbitrarily small neighborhood of their true values. Numerical simulations are used to demonstrate the effectiveness of the proposed estimation approaches.     

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower-upper symmetric Gauss-Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.     

Parallel simulation of concentration dynamics of nano-particles in High Gradient Magnetic Separation

In this paper, the concentration dynamics of weakly magnetic nano-particles dispersed in a fluid medium during the process of High Gradient Magnetic Separation is studied by using a computational approach. The continuity equations describing the time rate of change of the particle volume concentration in the system are solved numerically by using the finite difference method as initial and boundary value problems. Features of the concentration distribution in the system are depicted both in transient and steady states. Saturation buildup at the steady state and the time required for the occurrence of saturation buildup are given as basic guides for experiments. Parallel algorithms are implemented for simulations of concentration dynamics. The approaches are based on a distributed model, utilizing the message passing interface. Two schemes of parallel computing are developed based on different data-partitioning patterns and benchmark results are compared.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.