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UCFD_SPARSE
v1.0
Documentation
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UCFD_SPARSE
v1.0
Documentation
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Unstructured Grid Based CFD Applicable Asymmetric Sparse Matrix Numerical Library
UCFD_SPARSE provides LU-SGS (Lower Upper Symmetric Gauss-Seidel) type numerical methods for unstructured grid based Computational Fluid Dynamics (CFD) simulation. Since time accuracy is less important in steady flows problems, many researchers have introduced robust and fast-converging time marching schemes. The LU-SGS method is one of the widely-used methods proposed by Jameson & Yoon (1987) to obtain a steady solution, and has been successfully applied to the unstructured grid problem (Sharov & Nakahashi (1997)).
This library can be imported as a time integration process into CFD simulation code based on Finite Volume Method. Steady flow problems with Euler/Navier-Stokes/RANS equations are applicable, and both 2-dimensional and 3-dimensional grid are supported.
The main process called time integration using LU-SGS method consists of the following 4 steps.
1) Preparation : {serial/parallel}_pre_lusgs
In preparation phase, diagonal matrix of implicit operator is constructed. Diagonal matrix is slightly different with off-diagonal matrix, including time step and cell volume. Generally, inviscid flux Jacobian term is approximated with diffusive scheme like Rusanov flux, so that the diagonal matrix is reformulated by identity matrix with a scale factor. This replaces the matrix inversion process with simple scalar division, which reduces computation time dramatically.
2) Lower Sweep : {ns/rans}_{serial/parallel}_lower_sweep
After constructing diagonal matrix, lower and upper sweep are executed. Since lower sweep starts from least numbered cell to largest numbered cell, it is also called as Forward Sweep. Lower sweep results in the intermediate solution \(\Delta \vec{Q}^*\), which is stored in dub array. In this process, only lower neighbor cells already updated are required.
3) Upper Sweep : {ns/rans}_{serial/parallel}_upper_sweep
In contrast with the lower sweep, upper sweep explores cell reversely, from largest numbered cell to least numbered cell so that it is called as Backward Sweep. Colored LU-SGS also has the almost same procedure, reversing order of each cell cluster to compute. Upper sweep results in the difference between current and next time step solution \(\Delta \vec{Q}\). In this process, values in rhsb array are overwritten by the \(\Delta \vec{Q}\) because right-hand-side array is no longer needed when upper sweep executes. Similar to the lower sweep step, only upper neighbor cells already updated are required.
4) Update : {serial/parallel}_update
Current solution array is updated to the next time step solution. Solution of the next iteration step can be obtained by adding difference with the current solution. After updating, arrays to compute next time step solution (right-hand-side, time step, face spectral radius, etc) and total residual will be re-calculated. Process between update and preparation step depends on the types of which CFD solver is used, but generally second-order-accurate numerical method is applied into space discretization to capture shock wave well or get more precise flow results.
Generally, iterative time-stepping schemes like Block Jacobi, Gauss-Seidel method need sub-iteration steps to make solution difference array converge in current time step. However, sub-iteration steps can be removed when using LU-SGS method due to its remarkable robustness and stability.
RANS (Reynolds-Averaged Navier-Stokes) equations are supported by using integral formulation.
$$ \begin{align} \frac{\partial}{\partial t} \int_V {\vec{W}_T} \; dV + \oint_{\partial V} {(\vec{F}_{c_T} - \vec{F}_{v_T})} \; dS = \int_V {\vec{\mathbb{S}}_T \; dV} \end{align} $$
$$ \begin{align} \vec{W}_T & : \text{Conservative variables} \\ \vec{F}_{c_T} & : \text{Convective flux} \\ \vec{F}_{v_T} & : \text{Viscous flux} \\ \vec{\mathbb{S}}_T & : \text{Source term} \\ \end{align} $$
Turbulent. convective flux term based on Rusanov flux scheme Rusanov, V.V. (1962).coloredlusgs.c file in src folder.
1.9.5