A meshless and matrix-free fluid dynamics solver (SOMA) is introduced that avoids the need for user generated and/or analyzed grids, volumes, and meshes. Incremental building of the approximation avoids creation and inversion of possibly dense block diagonal matrices and significantly reduces user interaction. Validation results are presented from the application of SOMA to subsonic, compressible, and turbulent flow over an adiabatic flat plate.

The cost associated with Computational Fluid Dynamics (CFD) analysis has helped fuel the drive for programming methods that maximize the numerical data gathering capacity for a given level of computational effort and memory allocation. We believe that the Sequentially Optimized Meshfree Approximation (SOMA) method is another step in this direction [

SOMA is a meshless, volume-free, and matrix-free nonlinear differential equation solver. Flow simulations solvable by SOMA can be unsteady, compressible, and viscous. When flows are not solved purely steady state, SOMA has both explicit and implicit time stepping methods as well as the ability to switch dynamically between the two methods depending upon either computational efficiency or convergence criteria.

As a meshless method SOMA does not require Jacobians, information on mesh connectivity, or transformations between physical and computational domains through conformal maps. This simplifies the application of Adaptive Domain Refinement (ADR) techniques [

Other benefits and advancements associated with SOMA improve on both meshed and other meshless methods. Foremost among these, SOMA eliminates the need to invert large, possibly dense, block diagonal matrices in the solution of the governing equations. Solutions to the flow variables are determined through incrementally built approximations, and the governing equations can remain in vector form. Thus, SOMA eliminates the computational costs both of calculating and of inverting the system mass matrix.

The decades of analysis and work associated with other methods such as boundary element method (BEM), finite volume method (FVM) and finite difference method (FDM) with respect to numerical stability, upwinding, CFL number, characteristic lines, system eigenvalues, and many others can be directly applied to SOMA. This application can occur with little to no modification to account for the meshfree nature of SOMA. Additionally, boundary condition enforcement at farfield and surface locations can follow the methods used in FDM, FVM, and BEM [

A general approximation to some true function

where ^{th} basis function center in the problem domain.

In SOMA the approximations are built by holding the _{n−1} approximation fixed and optimizing the specific coefficient _{n} and basis function _{n}. The approximation can be written as

As there are no limitations on their distribution, a wide class of bases can be mixed and used (

For a differential equation, say

that can be reduced by growing the order ^{th} stage becomes

Reformulating this as an optimization problem, the objective function can be written as

where 〈 ·,· 〉 can be the sum root of the squares (

is used in this work, where _{p} is the number of evaluation points. The construction of the weighted series progresses until _{n} _{max}, where _{max} is a user defined accuracy threshold.

The basis function most often used in this work is the Gaussian Radial Basis Function (GRBF). The form for the ^{th} basis is

where _{i} is the scalar parameter that determines the width of the function. Thus, the GRBF must optimize ^{∞} continuous so derivatives can be determined directly.

However, capturing steep gradients in the dependent variable by simple smooth functions like GRBFs can be troublesome. To alleviate this difficulty, SOMA can augment the symmetric GRBF with an offset hyperbolic tangent function to create the “quasi-upwind” asymmetric Hyperbolic Radial Basis Functions (HRBF). These functions require additional parameters to determine the steepness of the asymmetry and its orientation but are also known analytically. The form for the ^{th} basis function is

where

SOMA optimizes both HRBF and GRBF using a genetic algorithm (GA) [_{n}, _{c,n}, _{n},

When the governing equations are coupled and/or multi-variable non-linear equations (

By default, GRBFs are used as they have fewer parameters to optimize. However, when a modeling problem requires too narrow a GRBF, SOMA switches to the HRBF form. We define a GRBF as too narrow when the radius encompasses fewer than five of the nearest evaluation points in two dimensions and nine points in three dimensions. The HRBF form is initialized as an asymmetric basis. Once the optimization routine no longer selects an asymmetric form (

This section explains the means by which derivatives are numerically calculated, the means for upwinding convective flows, and the means for switching between explicit and implicit time marching.

To start, our flux vector formulation of the Navier-Stokes equations is defined in the traditional manner

with repeated indices indicating summation. Using the three-dimensional Cartesian velocity vector

we can write

for the conserved variables density (_{c,i}) and energy (

for the velocity magnitude,

for the ideal gas pressure, and

for the viscous shear stress, where _{ij} is the Kronecker delta. We can now write

for the inviscid flux terms, and

for the viscous terms.

The working fluid in this work is considered to be air (specific heat ratio, _{∞} and _{∞} and the appropriate characteristic length

The normalized forms of energy and pressure used by SOMA were constructed such that they were unity at the farfield with a coefficient,

where _{∞} is the farfield Mach number. While this has no effect on the equations themselves, it reduces the optimization search space to a near unit hypercube, which has been shown to improve performance of SOMA in tests.

If we define the equation residuals for the mass, momentum, and energy equations as _{ρ}, _{Uc,j}, and _{E}, respectively, then the objective function for the Navier-Stokes validation problems is written as

where

for _{c },

Dirichlet boundary conditions are upheld directly by

Neumann or mixed conditions are enforced by using ghost points,

along the boundary _{N}.

For the special case of tangent/slip wall velocity (_{s,c} which converts from Cartesian coordinates to Surface Normal-Tangential-Binormal coordinates. The required velocity vector is

at each point along the surface boundary _{S}. The tangential/slip velocity boundary condition is enforced by setting

and the Cartesian velocity vector at the ghost points is calculated by inverting the rotation matrix _{s,c} against the Normal-Tangential-Binormal coordinate ghost velocity vector. Through the use of upwind techniques conditions can also be applied to arbitrary geometries.

Differential Quadrature (DQ) is used as the method for numerically calculating derivatives in this work. While it might seem that DQ would add extra computational cost to SOMA versus using exact function derivatives, it alleviates some of the major cost and difficulty associated with implementing boundary conditions on irregular geometries.

In this work the “node” is the point of interest and the remaining subset is referred to as “support points” or “supports”. ^{th}

where the node as a support point is included. The same formulation is used for derivatives of ^{y,m}_{ij} and ^{z,m}_{ij}, respectively. For an ^{th} ^{x,m−l,y,l}_{ij}.

Using a single independent variable as an example, the DQ weighting coefficients are determined using the known data from the basis functions. The function _{i},_{c,k}) and

where _{k}(_{c,k}. If

where

and _{x} is the matrix of

While it is possible to calculate higher order and cross derivatives with DQ, it is frequently more effective to calculate only first derivatives for the flow variables, combine them into flux vectors, and then calculate the derivatives of those. For example, with the three-dimensional Navier-Stokes equations, SOMA would need to calculate the first, second, and cross derivatives for all flow variables in all directions. For _{q} quadrature points in a domain of dimension ^{D−1} − 1) * _{q} * _{p}) operations while the flux formulation requires only _{q} * _{q} * _{p}) operations.

While typically _{q} is greater than 3, it is usually fewer than 20 meaning that in three dimensions the formulations are off by less than a factor of 7, and the extra work is offset by two major bonuses associated with the flux forms. First, there are fewer operations as there are fewer terms to multiply, and second, the flux forms are usually smoother than the primitive variables themselves [

In SOMA, DQ uses the Roe approximate third order solver [

Time marching techniques were used for the validation problems solved by SOMA even in cases which were physically at steady state. In these steady cases time marching will be referred to as false-transient methods. In all validation problems the two techniques available for time integration are explicit and implicit.

For explicit time (

Advantages of this Runge-Kutta method over the simple explicit Euler method [^{4}) compared to

Even accounting for the multi-step process of Runge-Kutta techniques, explicit methods are much less computationally expensive than implicit methods, but they also have the drawback of much smaller stable time steps. If the time step is bound by conditions other than stability (

In its most basic form, the implicit Euler method [

Because the updated flow variable is in both the unsteady and the nonlinear flux vector, the flux Jacobians [

Unlike these traditional methods there is no need for the inversion of matrices in SOMA which can be computationally expensive for three-dimensional external flows. With SOMA, the dependent variables are initialized as _{max}. The value of the approximation at the evaluation points is then saved and the previous weighted series is deleted. All dependent variables are updated and the new approximation is reinitialized as

An adaptive and local time step is applied in this work which gives the largest stable step possible for any given set of flow conditions. In the three-dimensional case, using the standard Courant-Friedrichs-Lewy parameter (CFL), the step is calculated as

with

from the Prandtl number (

and

are the inviscid and viscous eigenterms, respectively, of the system. We define _{j} as the acoustic speed at location

The inversion of a roughly sparse matrix with _{e} equations or dependent variables is _{q} and _{b} bases for each time step, the complexity is _{p}· _{e}· _{q}· _{b}) for SOMA. For three-dimensional external flows, this indicates that if we can maintain _{b} was highest for the initial condition.

SOMA typically solves the flow equations using an implicit formulation allowing for larger time steps and quick conversion to steady state or to stable periodic flows (_{implicit,i} = 95 _{explicit,i}.

If the optimization of _{max} when using the implicit formulation, the weighted series that was developed for the current time step is discarded. Initializing with the known solution from the previous time step, SOMA time marches explicitly until it has advanced the equivalent of one implicit time step. At this point the solver switches back to an implicit method and advances.

For a

and the heat flux term as

where _{t} and _{t} are the turbulent dynamic viscosity and thermal conductivity, respectively. Using the normalization described in

where _{t} is the turbulent Prandtl number.

In this work, we utilized the Menter single shear-stress-transport (SST)

and the governing equation for

With

which allows for time advancement of the remaining Navier-Stokes equations. Strong coupling of the SST model to the mass, momentum, and energy equations is straightforward.

For validation of the solution method, we chose relevant numerical problems of increasing complexity. SOMA was first used to approximate the solution to the steady state convective-diffusion equation (Burger’s equation) to test the method’s stability and convergence rate. Then SOMA was used to model vortex shedding about a circular cylinder (incompressible time-accurate Navier-Stokes equations) and on to inviscid, compressible flow past an NACA 0012 airfoil and ONERA M6 wing at angle of attack (compressible false-transient two- and three-dimensional Euler equations). Finally, SOMA was used to model the subsonic compressible turbulent flow over an adiabatic flat plate with zero pressure gradient. For the turbulence problem comparison data were taken from the NASA Langley Research Center flow verification website [_{max} was set to 1 × 10^{−7} for all results. The simulations were run in serial using a single-core Intel i5 processor. SOMA generally runs 10% slower on the validation problems as compared to our FVM solvers.

In this paper, the stationary convection-diffusion equation modeling a boundary layer also serves to evaluate the convergence of SOMA with increasing values of _{p}. The second order linear differential equation is given by

where

For our validation, the Reynolds number was set to 20 and SOMA was run up to _{p} = 300 with points uniformly distributed within the problem domain. The GRBF was utilized with three separate optimization routines: Fmincon, a pattern search method, and a GA [_{p}. The GA and pattern search give quintic convergence for the initial 20–50 GRBFs, then a linear convergence rate. The results for Fmincon are significantly worse both in terms of convergence and overall accuracy. Consequently, the GA was used in the remaining validation cases. It should be noted that all of the SOMA solutions for this case were stable for

Inlet conditions for this validation case were _{∞} = 0.5 and _{p} = 8,349 points. A comparison of the results is given in the form of time histories of the coefficients for both SOMA and reference [_{l} and _{d}) match well with the comparison data.

For this validation case the conditions used were angle of incidence _{∞} = 0.8. Results from a finite volume scheme [_{p} = 11,369 points. _{p}) along the top and bottom of the airfoil. Visual inspection of _{p} plots with respect to value distribution, shock locations, and lack of over- and undershoots at the shock locations. Quantitatively,

For this validation case the conditions used were _{∞} = 0.8358. Geometry specifics for the ONERA M6 wing are available in reference [_{P}) contours on the upper surface of the wing in _{p} = 72,791).

Method | _{upper}/ |
_{lower}/ |
---|---|---|

FVM | 0.610 | 0.32 |

SOMA | 0.608 | 0.32 |

In order to show the level of quantitative agreement, chord-wise distributions of the surface _{P} about the upper (U) and lower (L) surfaces of the wing are compared at span-wise locations _{P} data from reference [

The inflow conditions for the turbulent flow over a flat plate were _{L} = 5 × 10^{6}, which allows for calculating stagnation temperature and pressure values from isentropic relations. Extrapolation of static pressure to the boundary allows for calculation of the primitive variable values at the inlet. It should be noted that in _{L}, the

Results from the NASA test case were given using a 545 × 385 grid (equivalent to _{p} = 209,825). With the convergence of SOMA demonstrated in _{p} = 52,689 to demonstrate that SOMA can recover accurate results with a sparser discretization than reference [

Conventional techniques in numerical modeling can still require significant man hours for grid generation along with the need to invert large, sometimes dense, block diagonal matrices. With significant reduction in user interaction time the Sequentially Optimized Meshfree Approximation (SOMA) method serves as a new computational fluid dynamics solver eliminating the need for the previously mentioned matrices.

Unlike conventional techniques that require grids or volumes along with the corresponding connectivity data, SOMA only requires the coordinates of the evaluation points to model the flow. SOMA approximates the dependent variables incrementally through a greedy algorithm and minimization of the scalar form of the governing equations. As a result, the derivation and inversion of a condensed “mass” matrix are unnecessary. Reduction in the amount of user interaction time allows SOMA to improve as computing power continues to increase.

Although the three-dimensional validation case was restricted to the Euler equations, SOMA has been successfully applied to three-dimensional compressible laminar flow problems [

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.