Tetrahedral Grids Research Articles

RANS Simulation using the Spectral Volume Scheme on Unstructured Tetrahedral Grids

RANS Simulation using the Spectral Volume Scheme on Unstructured Tetrahedral Grids

Numerical Research on the Change Regularity of S Series Airfoil Tip Vortex

This paper take the impeller of S series airfoil for instance, in view of the complex flow field which is in the condition for the optimum attack angle of design wind velocity with tip vortex, presents a new generating strategy of grid based on controlling gradient method which takes the non-uniform tetrahedral grid with gradient gradual change in the near blade wall flow field. The transition layer grid which is in the development from body-fitted field to the middle of the domain having relatively high density of grid contrast with the far flow field can well capture the size and location of tip vortex, and find that the decay rate of the vorticity magnitude of tip vortex of S series impeller is more quickly than the tip vortex of traditional airfoil impeller. As well as find a fraction of central vortex and tip vortex shaded into each other in the far field and Doppler effect in the downstream. Because the numerical experiment and analysis showed that through controlling gradient change of grid can get high precision of calculating results, can also get S series impeller had higher efficiency in conversion of wind energy contrast with NACA series impeller, so the generating strategy of grid is reasonable and effective.

Four-dimensional radiotherapeutic dose calculation using biomechanical respiratory motion description

Organ motion due to patient breathing introduces a technical challenge for dosimetry and lung tumor treatment by hadron therapy. Accurate dose distribution estimation requires patient-specific information on tumor position, size, and shape as well as information regarding the material density and stopping power of the media along the beam path. A new 4D dosimetry method was developed, which can be coupled to any motion estimation method. As an illustration, the new method was implemented and tested with a biomechanical model and clinical data. First, an anatomical model of the lung and tumor was synthesized with deformable tetrahedral grids using computed tomography (CT) images. The CT attenuation values were estimated at the grid vertices. Respiratory motion was simulated biomechanically based on nonlinear finite element analysis. Contrary to classical image-based methods where motion is described using deformable image registration algorithms, the dose distribution was accumulated over tetrahedral meshes that are deformed using biomechanical modeling based on finite element analysis. The new method preserves the mass of the objects during simulation with an error between 1.6 and 3.6%. The new method was compared to an existing dose calculation method demonstrating significant differences between the two approaches and overall superior performance using the new method. A unified model of 4D radiotherapy respiratory effects was developed where biomechanical simulations are coupled with dose calculations. Promising results demonstrate that this approach has significant potential for the treatment for moving tumors.

Time domain finite volume method for three-dimensional structural–acoustic coupling analysis

This work extends the application of finite volume method (FVM) to structural–acoustic problems. A three-dimensional time domain FVM (TDFVM) is proposed to predict the transient response and natural characteristics of structural–acoustic coupling systems. Acoustic wave equation in heterogeneous medium and structural dynamic equation are solved in fluid and solid sub-domains respectively. The structural–acoustic coupling is implemented according to normal components of particle acceleration continuity condition and normal traction equilibrium condition at the interface. The computational domain is discretized with four-node tetrahedral grid which is generated easily and has strong adaptability to complicated geometries. Numerical experiments are carried out to examine the accuracy of the method in both time domain and frequency domain. The results show good agreement with analytical solutions and numerical results. For structural–acoustic problem, TDFVM has the capability to consider the heterogeneity of both fluid and solid.

Numerical analysis of non‐Newtonian blood flow and wall shear stress in realistic single, double and triple aorto‐coronary bypasses

Considering the fact that hemodynamics plays an important role in the patency and overall performance of implanted bypass grafts, this work presents a numerical investigation of pulsatile non-Newtonian blood flow in three different patient-specific aorto-coronary bypasses. The three bypass models are distinguished from each other by the number of distal side-to-side and end-to-side anastomoses and denoted as single, double and triple bypasses. The mathematical model in the form of time-dependent nonlinear system of incompressible Navier-Stokes equations is coupled with the Carreau-Yasuda model describing the shear-thinning property of human blood and numerically solved using the principle of the SIMPLE algorithm and cell-centred finite volume method formulated for hybrid unstructured tetrahedral grids. The numerical results computed for non-Newtonian and Newtonian blood flow in the three aorto-coronary bypasses are compared and analysed with emphasis placed on the distribution of cycle-averaged wall shear stress and oscillatory shear index. As shown in this study, the non-Newtonian blood flow in all of the considered bypass models does not significantly differ from the Newtonian one. Our observations further suggest that, especially in the case of sequential grafts, the resulting flow field and shear stimulation are strongly influenced by the diameter of the vessels involved in the bypassing.

Interactive Ray Casting of Geodesic Grids

AbstractGeodesic grids are commonly used to model the surface of a sphere and are widely applied in numerical simulations of geoscience applications. These applications range from biodiversity, to climate change and to ocean circulation. Direct volume rendering of scalar fields defined on a geodesic grid facilitates scientists in visually understanding their large scale data. Previous solutions requiring to first transform the geodesic grid into another grid structure (e.g., hexahedral or tetrahedral grid) for using graphics hardware are not acceptable for large data, because such approaches incur significant computing and storage overhead. In this paper, we present a new method for efficient ray casting of geodesic girds by leveraging the power of Graphics Processing Units (GPUs). A geodesic grid can be directly fetched from storage or streamed from simulations to the rendering stage without the need of any intermediate grid transformation. We have designed and implemented a new analytic scheme to efficiently perform value interpolation for ray integration and gradient calculations for lighting. This scheme offers a more cost‐effective rendering solution over the existing direct rendering approach. We demonstrate the effectiveness of our rendering solution using real‐world geoscience data.

Forward modeling of gravity data using finite-volume and finite-element methods on unstructured grids

Minimum-structure inversion is one of the most effective tools for the inversion of gravity data. However, the standard Gauss-Newton algorithms that are commonly used for the minimization procedure and that employ forward solvers based on analytic formulas require large memory storage for the formation and inversion of the involved matrices. An alternative to the analytical solvers are numerical ones that result in sparse matrices. This sparsity suits gradient-based minimization methods that avoid the explicit formation of the inversion matrices and that solve the system of equations using memory-efficient iterative techniques. We have developed several numerical schemes for the forward modeling of gravity data using the finite-element and finite-volume methods on unstructured grids. In the finite-volume method, a Delaunay tetrahedral grid and its dual Voronoï grid are used to find the primary solution (i.e., gravitational potential) at the centers and vertices of the tetrahedra, respectively (cell-centered and vertex-centered schemes). In the finite-element method, Delaunay tetrahedral grids are used to develop linear and quadratic finite-element schemes. Different techniques are used to recover the vertical component of gravitational acceleration from the gravitational potential. In the finite-volume scheme, a differencing method is used; in the finite-element method, basis functions are used. The capabilities of the finite-volume and finite-element schemes were tested on simple and realistic synthetic examples. The results showed that the quadratic finite-element scheme is the most accurate but also the most computationally demanding scheme. The best trade-offs between accuracy and computational resource requirement were achieved by the linear finite-element and vertex-centered finite-volume schemes.

A large-scale nonlinear eigensolver for the analysis of dispersive nanostructures

We introduce the electromagnetic eigenmodal solver code FemaxxNano for the numerical analysis of nanometer structured optical systems, a scientific field generally know as nanooptics. FemaxxNano solves the electric field vector wave equation and calculates the electromagnetic eigenmodes of nearly arbitrary 3-dimensional resonators, embedded either in free-space, vacuum or a background medium. Here, the study of the interaction between nanometer sized metallic structures and light is at the heart of the physical problem. Since metals in the optical region of the electromagnetic spectrum are highly dispersive and, thus, dissipative, dielectric media, we eventually obtain a nonlinear eigenvalue problem. We discretize the electromagnetic eigenvalue problem with the finite element method (FEM) in 3-dimensional space and on unstructured tetrahedral grids. We introduce a fully iterative scheme to solve the nonlinear problem for complex coefficient matrices that depend on wavelength. We investigate the properties of the algorithm in detail and demonstrate its performance by analyzing a nanometer sized optical dimer structure, a specific type of optical antenna, on distributed-memory parallel computers.

Regional wave propagation using the discontinuous Galerkin method

Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

A 3D hybrid grid generation technique and a multigrid/parallel algorithm based on anisotropic agglomeration approach

A hybrid grid generation technique and a multigrid/parallel algorithm are presented in this paper for turbulence flow simulations over three-dimensional (3D) complex geometries. The hybrid grid generation technique is based on an agglomeration method of anisotropic tetrahedrons. Firstly, the complex computational domain is covered by pure tetrahedral grids, in which anisotropic tetrahedrons are adopted to discrete the boundary layer and isotropic tetrahedrons in the outer field. Then, the anisotropic tetrahedrons in the boundary layer are agglomerated to generate prismatic grids. The agglomeration method can improve the grid quality in boundary layer and reduce the grid quantity to enhance the numerical accuracy and efficiency. In order to accelerate the convergence history, a multigrid/parallel algorithm is developed also based on anisotropic agglomeration approach. The numerical results demonstrate the excellent accelerating capability of this multigrid method.

An unstructured, three-dimensional, shock-fitting solver for hypersonic flows

A novel unstructured shock-fitting algorithm for three-dimensional flows is proposed in this paper. This new technique is able to work coupled with any unstructured vertex centred solver. The fitted shock front is described using a triangulated surface that is allowed to move, while obeying to the Rankine–Hugoniot jump relations, throughout a background tetrahedral mesh which covers the entire computational domain. At each time step, a local, constrained Delaunay tetrahedralization is applied in the neighbourhood of the shock front to ensure that the triangles, which make up the shock surface, belong to the overall tetrahedral grid. The fitted shock front acts as an interior boundary for the unstructured shock-capturing solver, used to solve the discretised governing equations in the smooth regions of the flow-field. The present algorithm has been tested against high speed flows past three-dimensional bodies, providing high quality results even using coarse grain tetrahedralization.

Large-Eddy Simulation of Heat Transfer Around a Square Cylinder Using Unstructured Grids

This paper presents a method of large-eddy simulation on unstructured grids designed to predict the wall heat transfer in typical aeronautical applications featuring turbulent flows and complex geometries. Two types of wall treatments are considered: a wall-function model using a full tetrahedral grid and a wall-resolved method computed on a hybrid tetrahedral–prismatic grid. These two approaches are tested against the square-cylinder case at moderate Reynolds number (), in which many reference data are available for flow dynamics and heat transfer. Both predict accurately the unsteady flow around the cylinder and in its near wake, but only the wall-resolved approach reproduces the Nusselt-number global value and its spatial distribution around the cylinder wall. This latter method is used to investigate the coupling between periodic vortex shedding and wall heat transfer using a phase-averaged analysis.

Optimization of the multigrid-convergence rate on semi-structured meshes by local Fourier analysis

In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother is presented as an efficient choice for regular tetrahedral grids, whereas line and plane relaxations are needed for poorly shaped tetrahedra. A novel partitioning of the Fourier space is proposed to analyze the four-color smoother. Numerical test calculations validate the theoretical predictions. A multigrid method is constructed in a block-wise form, by using different smoothers and different numbers of pre- and post-smoothing steps in each tetrahedron of the coarsest grid of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

Method of adaptive artificial viscosity for gas dynamics equations on triangular and tetrahedral grids

In studies [1–7] a method of adaptive artificial viscosity (AAV) was proposed for the solution of gas dynamics equations. In this paper, this method is extended to the case of triangular grids for two-dimensional (2D) equations in the variables x, y, and r, z, and of tetrahedral grids for equations in Cartesian variables x, y, and z. The calculation results for the test problems are presented.

Numerical Prediction of Surface Heat Flux During Multiple Jets Firing for Missile Control

Numerical simulations are carried out to obtain the flowfield and heat flux arising out of the flow interactions of different control thrusters viz. vernier, pitch, yaw, roll and divert thruster among themselves as well as with free stream at different altitudes of operation. Three critical points on a typical trajectory of a missile are chosen and combinations of the thrusters operating at those conditions are considered. Simulations have also been performed to simulate DT motor interaction with free stream at different altitude. The interaction of different motor flowfield with the free stream presents a very complex flowfield. Flow gradients are very high close to the nozzle exit because of high altitude operation of motors. 3-Dimensional RANS equations are solved along with k-e turbulence model on unstructured tetrahedral grid using commercial CFD software. The flow properties along with the surface heat flux distribution for four isothermal wall temperatures 350, 450, 550, 650 K are computed and provided for surface temperature prediction. It is observed that with increase in altitude, the high heat flux region reduces and heat transfer coefficient is independent of wall temperature.

A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids

A reconstructed discontinuous Galerkin (RDG) method based on a hierarchical WENO reconstruction, termed HWENO (P1P2) in this paper, designed not only to enhance the accuracy of discontinuous Galerkin methods but also to ensure the nonlinear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this HWENO (P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a Hermite WENO reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the RDG method. The developed HWENO (P1P2) method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, robustness, and non-oscillatory property. The numerical experiments indicate that the HWENO (P1P2) method is able to capture shock waves within one cell without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method.

Simulation of flow and heat transfer in a liquid drop sliding underneath a hydrophobic surface

Clusters of liquid drops growing and moving on physically or chemically textured surfaces are encountered in dropwise mode of vapor condensation. This process can be sustained only if the surface integrity is maintained over a long period of time. Surface features are altered when sliding drops leach away the promoter layer. In the absence of chemical reactions between the promoter and the condensing liquid, the wall shear stress is the primary parameter controlling the physical leaching. In turn, wall shear stress depends on the relative speed between the drop and the substrate surface and the shape of the drop. Given a wall shear stress distribution for individual drops, the net effect of an ensemble of them during continuous quasi-steady state dropwise condensation can be determined using the population density of drops.Wall shear stress and local heat fluxes have been determined in the present work by solving the Navier–Stokes and energy equations in three-dimensions on an unstructured tetrahedral grid that represents an individual droplet. The drop size and relative velocity are parameterized by the Reynolds number (Re=10–1000), apparent contact angle (90–120°) and its shape. The simulations presented here are for a wide range of Prandtl numbers, i.e., 0.005–30. The wall shear stress and wall heat flux are expressed in terms of the skin friction coefficient and the Nusselt number, respectively. While these two quantities show an increase with Reynolds number, they decrease at higher values of the drop contact angle on/underneath hydrophobic surface. At low Prandtl numbers, heat transfer is mainly diffusional and the wall Nusselt number is practically independent of Reynolds number at any given Pr. The maximum wall shear stress as well as heat flux occurs at the corners of the drop close of the three-phase contact line. The surface averaged shear stress and heat flux are expressed in terms of appropriate correlations that include Reynolds number, Prandtl number, and the apparent contact angle. The wall shear stress in the relatively inactive central region at the drop base is smaller than the overall base average by a factor of 6, while that for heat transfer, the corresponding factor is in the range of 1.3–1.8. The figure of merit function, represented by the ratio of average Nusselt number to the friction coefficient, increases with contact angle, indicating an advantage to be gained from hydrophobic surfaces. The information presented in this paper is vital for further improvement of the available models of dropwise condensation on textured surfaces.

Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.

3D numerical analysis of flow in Francis turbine in sandy river

Three-dimensional turbulent flow calculation of the flow in Francis turbine in sandy river was conducted by using body-fitted coordinates, tetrahedral grid system and SIMPLE algorithm based on the solid-liquid two-phase flow equation of motion in Eulerian coordinate system. The flow mechanism of sandy water in the turbine was analyzed through numerical simulation of flow in the turbine in several different sand water concentrations. In sandy water with various concentrations, the erosion area of turbine are basically concentrated at the lateral wall of the whole flow passage and the heads of guide vanes and blades.

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.