Flow simulations in arbitrarily complex cardiovascular anatomies – An unstructured Cartesian grid approach

Image guided computational fluid dynamics is attracting increasing attention as a tool for refining in vivo flow measurements or predicting the outcome of different surgical scenarios. Sharp interface Cartesian/Immersed-Boundary methods constitute an attractive option for handling complex in vivo geometries but their capability to carry out fine-mesh simulations in the branching, multi-vessel configurations typically encountered in cardiovascular anatomies or pulmonary airways has yet to be demonstrated. A major computational challenge stems from the fact that when such a complex geometry is immersed in a rectangular Cartesian box the excessively large number of grid nodes in the exterior of the flow domain imposes an unnecessary burden on both memory and computational overhead of the Cartesian solver without enhancing the numerical resolution in the region of interest. For many anatomies, this added burden could be large enough to render comprehensive mesh refinement studies impossible. To remedy this situation, we recast the original structured Cartesian formulation of Gilmanov and Sotiropoulos [Gilmanov A, Sotiropoulos F. A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J Comput Phys 2005;207(2):457–92] into an unstructured Cartesian grid layout. This simple yet powerful approach retains the simplicity and computational efficiency of a Cartesian grid solver, while drastically reducing its memory footprint. The method is applied to carry out systematic mesh refinement studies for several internal flow problems ranging in complexity from flow in a 90° pipe bend to flow in an actual, patient-specific anatomy reconstructed from magnetic resonance images. Finally, we tackle the challenging clinical scenario of a single-ventricle patient with severe arterio-venous malformations, seeking to provide a fluid dynamics prospective on a clinical problem and suggestions for procedure improvements. Results from these simulations demonstrate very complex cardiovascular flow dynamics and underscore the need for high-resolution simulations prior to drawing any clinical recommendations.     

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.     

An improved immersed boundary method for curvilinear grids

In the present paper we propose an extension of the direct-forcing immersed boundary technique, recently developed and employed by Verzicco and co-authors [Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J Comput Phys 2000;161:35–60; Verzicco R, Fatica M, Iaccarino G, Moin P, Khalighi B. Large eddy simulation of a road vehicle with drag-reduction devices. AIAA J 2002;40(12):2447–55; Cristallo A, Verzicco R. Combined immersed boundary/large-eddy-simulations of incompressible three-dimensional complex flows. Flow Turbul Combust 2006;77(1–4):3–26.] and successively improved by Balaras and co-authors [Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. J Comput Phys 2003;191:660–9; Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput Fluids 2004;33:375–404]. We extend the aforementioned technique to curvilinear-coordinate, structured grid, Navier–Stokes solvers. This improved technique allows for more flexibility and efficiency when compared to standard methods in which the technique is coupled with orthogonal-grid solvers. Additional modifications are also proposed with respect to the state-of-art, which allow to deal with general shaped, multiple-body immersed surfaces and to make the interpolation of the velocity field off the body suitable for curvilinear grids. Several tests have been carried out to check the reliability of the proposed technique: first we have considered the three-dimensional Stokes flow around a sphere, and compared the numerical results with the analytical ones. Second we have considered the two-dimensional unsteady flow around a circular cylinder placed between two parallel solid walls and compared the results with those of the database of the Priority Research Program ‘Flow Simulation on High Performance Computers’ of the German Research Association (DFG). Third, we have considered the two-dimensional flow within a S-shaped duct containing an elliptical valve. Finally, we have applied the technique to the study of a practical high-Reynolds number industrial problem.The geometrical configuration of the first two test cases is suited for both Cartesian and curvilinear algorithms. The geometry of the third test case is suited for curvilinear meshes and makes the use of Cartesian grids very inefficient and less accurate than the curvilinear ones. In these cases Cartesian – as well as curvilinear – mesh simulations have been carried out. Finally, the geometry of the high-Reynolds industrial problem is suited for curvilinear grids.The proposed technique has shown to preserve at least the same level of accuracy of its Cartesian counterpart allowing to reduce in a considerable way the computational cost of the simulations. When the geometry is better suited for curvilinear meshes, the reduction of the computational cost is accompanied by an increased accuracy with respect to the Cartesian counterpart.We also propose a simplified direct-forcing, semi-implicit method, allowing reduced computational cost with respect to the literature techniques. We have checked the accuracy of the technique and shown that when the Reynolds number is large enough, the present simplified technique allows the use of time steps much larger than those allowed by the explicit time-advancement scheme, preserving the accuracy of the results.     

A parallel boundary fitted dynamic mesh solver for applications to flapping flight

A parallel numerical solution procedure for unsteady incompressible flow is developed for simulating the dynamics of flapping flight. A collocated finite volume multiblock approach in a general curvilinear coordinate is used with Cartesian velocities and pressure as dependent variables. The Navier-Stokes equations are solved using a fractional-step algorithm. The dynamic grid algorithm is implemented by satisfying the space conservation law by computing the grid velocities in terms of the volume swept by the faces. The dynamic movement of grid in a multiblock approach is achieved by using a combination of spring analogy and Trans-Finite Interpolation. The spring analogy is used to compute the displacement of block corners, after which Trans-Finite Interpolation is applied independently on each computational block. The performance of the code is validated in forced transverse oscillations of a cylinder in cross-flow, a heaving airfoil, and hovering of a fruitfly. Finally, the unsteady aerodynamics of flapping flight at Re = 10,000 relevant to the development of Micro Air Vehicles is analyzed for forward flight. The results show the capability of the solver in predicting unsteady aerodynamics characterized by complex boundary movements.     

A novel immersed boundary procedure for flow and heat simulations with moving boundary

This article describes a novel immersed boundary procedure for computing the flow and heat transfer problems with moving and complex boundary. Although the immersed boundary techniques have been successfully implemented to these flow and heat simulations, a frequently encountered drawback of this method is the relatively low accuracy proximate to the boundary due to the spreading of forcing function or the interpolation scheme. In this study, we propose a moving-grid process under the arbitrary Lagrangian-Eulerian framework to reduce the numerical diffusion near the immersed boundary. The incompressible Navier-Stokes equations are discretized spatially using unstructured finite element method, and advanced temporally by an operator-splitting scheme. The methodology is validated by the simulations of flow induced by an oscillating cylinder in a free stream. The capability of the proposed method is further demonstrated by good predictions of flow passing the rotating fan in a channel and also flow driven by two independent rotating fans in a circular cavity.     

A calculation procedure for turbulent flow in complex geometries

This paper provides a complete exposition of a general finite-volume approach to the calculation of turbulent flow in geometrically complex domains, based on a non-orthogonal cordinate formulation of the governing equations with contravariant physical velocity components. It commences with a useful coordinate transformation which allows the aforementioned equations to be easily derived from the Cartesian coordinate counterparts. Then an implicit finite volume method of solution is presented, together with a simple transfinite mapping procedure for mesh generation. Finally the overall method is applied, by way of an example, to the prediction of flow and heat transfer in a staggered tube bank.     

Block Cartesian abstraction of a geometric model and its application in hexahedral mesh generation

In this work, a fuzzy logic approach is proposed to transform a geometric model of arbitrary shape to its block Cartesian abstraction. This abstraction is topologically similar to the original model and it contains geometric sub-entities which are all aligned in the Cartesian directions. This is achieved by calculating the modifications made to the face normal vectors as a result of the influences of the adjacent faces. A fuzzy logic inference engine is developed by combining heuristics to emulate the local changes in face normal vectors with respect to the changes in the global space. A three-dimensional field morphing algorithm is used to position the features of this block Cartesian abstraction so that a congruent geometric model can be reconstructed. Such a model is useful for the generation of structured quadrilateral boundary element meshes or structured hexahedral meshes based on grid-based meshing method, mesh mapping or sweeping. This approach is also able to overcome the traditional problem of having poorly shaped elements at the boundary using the grid-based method of mesh generation. As the topology of the block Cartesian abstraction is congruent to the original model, the mesh can be mapped back to the original model by employing an inverse operation of the transformation.     

An immersed interface method for the Vortex-In-Cell algorithm

The paper presents a two-dimensional immersed interface technique for the Vortex-In-Cell (VIC) method for simulation of flows past bodies of complex geometry. The particle–mesh VIC algorithm is augmented by a local particle–particle correction term in a Particle–Particle Particle–Mesh (P³M) context to resolve sub-grid scales incurred by the presence of the immersed interface. The particle–particle correction furthermore allows to disjoin mesh and particle resolution by explicitly resolving sub-grid scales on the particles. This P³M algorithm uses an influence matrix technique to annihilate the anisotropic sub-grid scales and adds an exact particle–particle correction term. Free-space boundary conditions are satisfied through the use of modified Green’s functions in the solution of the Poisson equation for the streamfunction. The concept is extended such as to provide exact velocity predictions on the mesh with free-space boundary conditions.The random walk technique is employed for the diffusion in order to relax the need for a remeshing of the computational elements close to solid boundaries. A novel partial remeshing technique is introduced which only performs remeshing of the vortex elements which are located sufficiently distant from the immersed interfaces, thus maintaining a sufficient spatial representation of the vorticity field.Convergence of the present P³M algorithm is demonstrated for a circular patch of vorticity. The immersed interface technique is applied to the flow past a circular cylinder at a Reynolds number of 3000 and the convergence of the method is demonstrated by a systematic refinement of the spatial parameters. Finally, the flow past a cactus-like geometry is considered to demonstrate the efficient handling of complex bluff body geometries. The simulations offer an insight into physically interesting flow behavior involving a temporarily negative total drag force on the section.     

Quadtree grid generation: Information handling, boundary fitting and CFD applications

When solving partial differential equations by numerical methods, an automatic mesh generation technique which can accommodate local mesh refinement adaptively is desirable. One efficient technique for producing such meshes in two-dimensional space is to subdivide recursively the domain into quadrants using a quadtree to store and manipulate the mesh information. Here, the quadtree grid generation technique is reviewed and its programming discussed. Three data storage methods are examined. The conversion of the quadtree grid to a triangular finite element mesh is also described, along with methods for fitting the mesh to smooth boundary contours. Results from viscous flow and standing wave simulations are used to illustrate mesh adaptivity about internal and boundary features.     

Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method

This paper presents a lattice Boltzmann method (LBM) based study aimed at numerical simulation of highly turbulent and largely inclined flow around obstacles of curved geometry using non-body-fitted Cartesian meshes. The approach features (1) combining the interpolated bounce-back scheme with the LBM of multi-relaxation-time (MRT) type to enable the use of simple Cartesian mesh for the flow cases even with complex geometries; and (2) incorporating the Spalart–Allmaras (SA) turbulence model into LBM in order to represent the turbulent flow effect. The numerical experiments are performed corresponding to flows around an NACA0012 airfoil at Re=5×10⁵ and around a flat plate at Re=2×10⁴, respectively. The agreement between all simulation results obtained from this study and the data provided by other literature demonstrates the reliability of the enhanced LBM proposed in this paper for simulating, simply on Cartesian meshes, complex flows that may involve bodies of curved boundary, high Reynolds number, and large angle of attack.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

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.     

Coupling geological and numerical models to simulate groundwater flow and contaminant transport in fractured media

A new modeling approach is presented to improve numerical simulations of groundwater flow and contaminant transport in fractured geological media. The approach couples geological and numerical models through an intermediate mesh generation phase. As a first step, a platform for 3D geological modeling is used to represent fractures as 2D surfaces with arbitrary shape and orientation in 3D space. The advantage of the geological modeling platform is that 2D triangulated fracture surfaces are modeled and visualized before building a 3D mesh. The triangulated fractures are then transferred to the mesh generation software that discretizes the 3D simulation domain with tetrahedral elements. The 2D triangular fracture elements do not cut through the 3D tetrahedral elements, but they rather form interfaces with them. The tetrahedral mesh is then used for 3D groundwater flow and contaminant transport simulations in discretely fractured porous media. The resulting mesh for the 2D fractures and 3D rock matrix is checked to ensure that there are no negative transmissibilities in the discretized flow and transport equation, to avoid unrealistic results. To test the validity of the approach, flow and transport simulations for a tetrahedral mesh are compared to simulations using a block-based mesh and with results of an analytical solution. The fluid conductance matrix for the tetrahedral mesh is also analyzed and compared with known matrix values.     

Ghost-Cell Method with far-field coarsening and mesh adaptation for Cartesian grids

Cartesian grid methods for inviscid computational fluid dynamics offer great promise for the development of very rapid conceptual design tools. The present paper deals with a number of new features for Cartesian grid methods which appear to be particularly well suited for this application. A key ingredient is the implementation of non-penetration boundary conditions at solid walls which is based upon the curvature-corrected symmetry technique (CCST) developed by the present authors for body-fitted grids. The method introduces ghost cells near the boundaries whose values are developed from an assumed flow-field model in vicinity of the wall. This method was shown to be substantially more accurate than traditional surface boundary condition approaches. This improved boundary condition has been adapted to a Cartesian mesh formulation, which we termed the Ghost-Cell Method. In this approach, all cell centers exterior to the body are computed with fluxes at the four surrounding cell edges, without any cut cells which complicate other Cartesian mesh methods. Another typical drawback of non-adaptive Cartesian grid methods is that any Cartesian grid clustering near the body must be maintained to the far field boundary. To address this issue, we have introduced a far-field grid coarsening, based on the iblanking approach, which maintains the structured nature of the grid while computing only the active cell centers. In addition, to highlight the advantages of grid adaptation in connection with Cartesian mesh methods, we have introduced a rudimentary procedure which detects the shock position and automatically refines the mesh by locally reducing the grid dimension to one fourth of its original dimension. The merits of the Ghost-Cell Method are established by the computation of the compressible flow about circular cylinders. The results show the surface non-penetration condition to be satisfied in the limit of vanishing cell size and the method to be second-order accurate in space. The results of the far-field grid coarsening indicate that the numerical solutions are unaffected by the coarsening, while the number of the computed cells and the CPU time are reduced to less than 50% of the uncoarsened solution values. The results computed using the mesh adaptation at the shock indicate that the method is effective in reducing the shock transition region thickness, without modifying the flow solution away from the shock. Finally, some test cases with airfoils located at different positions have been considered and the results are proven to be practically independent of the position of the body with respect to the grid. Although this paper is limited to two-dimensional applications, the methodology is well-suited for general three-dimensional geometries, which will be the subject of future research.     

Free-boundary method for the numerical solution of gas-dynamic equations in domains with varying geometry

A novel method is proposed for numerical solution of gas-dynamic equations on stationary Cartesian grids in domains containing solid impermeable and, in the general case, moving inclusions (objects). The suggested technique is based on the immersed boundary method, in which the computational domain (including solid objects) is covered by a single Cartesian grid and the calculation is carried out by the shock-capturing method over all cells. Under this approach, the influence of the solid inclusions on the flow of the gas medium is simulated by the introduction of specially selected mass, momentum, and energy fluxes into the right-hand side of the equations. The currently developed methods for the solution of this class of problems are surveyed and the advantages of the proposed approach are discussed. The method is verified by calculating some test problems that admit analytical solutions and it is used to solve the problem of supersonic flow around a blunt body. The results are compared with the calculation findings based on the standard curvilinear grid tied to the geometry of the body.     

Structural shape optimization using Cartesian grids and automatic <Emphasis Type="Italic">h</Emphasis>-adaptive mesh projection

We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of the matrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h-adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process.     

Adaptive boundary layer meshing for viscous flow simulations

A procedure for anisotropic mesh adaptation accounting for mixed element types and boundary layer meshes is presented. The method allows to automatically construct meshes on domains of interest to accurately and efficiently compute key flow quantities, especially near wall quantities like wall shear stress. The new adaptive approach uses local mesh modification procedures in a manner that maintains layered and graded elements near the walls, which are popularly known as boundary layer or semi-structured meshes, with highly anisotropic elements of mixed topologies. The technique developed is well suited for viscous flow applications where exact knowledge of the mesh resolution over the computational domain required to accurately resolve flow features of interest is unknown a priori. We apply the method to two types of problem cases; the first type, which lies in the field of hemodynamics, involves pulsatile flow in blood vessels including a porcine aorta case with a stenosis bypassed by a graft whereas the other involves high-speed flow through a double throat nozzle encountered in the field of aerodynamics.     

混合重叠网格通信优化算法

重叠网格技术广泛应用在复杂外型和运动边界问题的流场数值模拟中.本文在并行重叠网格隐式挖洞算法实现的基础上,提出了笛卡尔辅助网格和多块结构网格的混合重叠网格方法.通过笛卡尔辅助网格实现重叠网格洞边界和网格插值关系的快速建立.通过定义重叠区域网格权重、部件网格与背景网格绑定的方法,建立了混合网格的并行分配模式,有效减少重叠插值信息在各进程间的通信,实现计算负载和通信负载在各个进程的均匀分配.测试表明该方法可应用于数千万量级的重叠网格系统,可扩展至千核规模,高效的实现多个物体构成的复杂网格系统的重叠关系建立.