ACUSIM Software has been acquired by Altair Engineering and the AcuSolve software is now part of Altair's HyperWorks product line. Click here to obtain latest information about AcuSolve.


The flow solver is based on the Galerkin/Least-Squares (GLS) finite element technology developed at Stanford University during late 80's and early 90's. This technology has since been enhanced by ACUSIM to better handle large-scale industrial problems.

Finite Element Formulation

The method is based on the finite element Galerkin weighted residual formulation with equal-order interpolation for all solution fields, including pressure. This not only has the advantage of simplicity of coding and integration into scientific and engineering applications, but also is crucial for retaining the accuracy of the underlying scheme.

A least-squares operator is added to the basic method to achieve stability of the divergence-free constraint and the convective terms. Discontinuity-capturing and nonlinear maximum principal operators are added to resolve sharp internal and boundary discontinuities and oscillations that are not resolved by the (linear) least-squares operator. These operators are mathematically designed to yield the needed stability without sacrificing the underlying accuracy and conservative nature of the Galerkin formulation. These operators can be contrasted to artificial diffusion operators adopted by many commercial CFD packages, where stability is achieved at the expense of accuracy and/or conservation.

The method is also designed to maintain local and global conservation of the differential equations. The conservation is achieved for any meshes.

In addition to excellent spatial accuracy, AcuSolve has a second-order time accurate option. When combined with the coupled pressure/velocity linear solver (see Robustness), rapid nonlinear convergence in each time step is obtained. This leads to the realization of the second-order time-accurate solutions. This may be contrasted with segregated type solution schemes used by many commercial CFD packages, where converging the nonlinear iteration at each time step is typically not feasible, and time accuracy is rarely observed.

The resulting technology has a very rich mathematical foundation. In practice, it exhibits a high degree of robustness, it is very accurate, and always conservative.

Backward Facing Step Problem

Solution of the turbulent flow over a backward-facing step at Reynolds number of 40,000 is used to demonstrate the accuracy of the method. The mesh consists of one slice of 3D elements, having 7K brick elements and 15K nodes. The Spalart-Allmaras turbulence model is used here. The reattachment length is an excellent measure of solution accuracy. The computed reattachment length is 7.05 times the step height, which is in excellent agreement with the experimental results of 7. In addition the figure below shows a number of particle paths in the separation region at the step. The mesh is superimposed on the figure for reference. The method has captured not only the main separation eddy, but also two secondary eddies at the corner. The smallest eddy is captured within a radius of three elements. This strongly attests to the accuracy of the methods.

Conservation Example

The flow over a backward-facing step is used to demonstrate the conservative nature of the method. An arbitrary cut (on element boundaries) is made in the interior of the mesh through the main separation eddy. With this cut, we have a closed loop consisting of Inflow, Top Section, Bottom Section and Interior Cut.

The problem is converged to 10-5. The table below shows the mass flux and each of the momentum fluxes across each boundary. Note that the total fluxes add up to zero.

Surface Mass Flux X-Momentum Y-Momentum
Advection Cauchy Advection Cauchy
Top Section
Bottom Section
Interior Cut
Total -0.01 3.02 -3.02 3.46 -3.46
0 0