**Solution Methods**

**Overview** • Properties of Numerical Solution Methods • FVM and FEM solution methods • Characteristics of solution algorithms • Equations solvers • Underrelaxation • Convergence

**Numerical Solution Methods (1)** • The important components of a numerical solution method are: 1. Mathematical model of flow • e.g. equations of motion- unsteady and steady, compressible and incompressible, 2D and 3D, turbulence, etc. 2. Discretization Method • Approximation of the differential equations by a system of algebraic equations • Finite Difference Method (FDM) • Finite Volume Method (FVM) • Finite Element Method (FEM) 3. Coordinate system • cartesian or cylindrical, curvilinear orthogonal and non-orthogonal coordinate systems

**Numerical Solution Methods (2)** 4. Numerical Grid • The solution domain is subdivided by the grid. The algebraic conservation equations for the variables are computed on a finite number of control volumes or elements in the domain. • Types of Grids • Structured grids • Multi-block-structured grids • Unstructured grids Unstructured surface grid for vehicle aerodynamic analysis. 5. Finite Approximations • Discretizing the solution domain gives rise to errors from the approximation of the continuous differential functions • FDM - approximate the derivatives through the Taylor series expansion • FVM - approximate the surface and volume integrals • FEM - choose weighting functions

**Numerical Solution Methods (3)** 6. Solution Criteria and Convergence Criteria • This is the topic of this lecture • Methods of solving the system of algebraic equations • The nonlinear nature of the governing equations requires an iterative solution method. Convergence criteria determine when to terminate the iterative process. Accuracy and efficiency are considered.

**Properties of Solution Methods** • Consistency • Stability • Convergence • Conservation • Boundedness • Realizability • Accuracy

**N** n j E P W e w s i S FVM - Solution Algorithms • The discretized form of the governing conservation equations can be written as: • where nb denotes the cell neighbors of cell P • In a 2D structured grid, the face P has fourneighbors (E,W,N,S). In a 3D grid, a cell hassix neighbors. • In an unstructured grid, the number of neighbors depends on the cell shape and mesh topology. • The above algebraic equation is written for each transport variable, that is, velocity, temperature, species concentration and turbulence quantities.

**FVM - Solution Algorithms** • The solution of the Navier-Stokes equations is complicated by the lack of an independent equation for pressure. Pressure is linked to all three momentum equations • The pressure-velocity coupling algorithm SIMPLE (Semi-Implicit Pressure Linked Equations), and it’s variants, are used. • Concept: • the momentum equations are used to compute velocity • a pressure equation is derived from the continuity equation • a discrete pressure correction equation is derived from the discrete forms of the pressure and momentum equations • the pressure correction equation is updated with pressure and a mass flux balance through a mass correction

**Finite Volume Solution Methods** • The Finite Volume Solution method can either use a “segregated” or a “coupled” solution procedure. • The solution procedure of each method is the same.

**Update properties.** Solve momentum equations (u, v, w velocity). Solve pressure-correction (continuity) equation. Update pressure, face mass flow rate. Solve energy, species, turbulence, and other scalar equations. Converged? No Yes Stop Segregated Solution Procedure

**No** Yes Coupled Solution Procedure Update properties. Solve continuity, momentum, energy, and species equations simultaneously. Solve turbulence and other scalar equations. Converged? Stop

**No** Yes Stop Unsteady Solution Procedure • Same procedure for segregated and coupled solvers: Execute segregated or coupled procedure, iterating to convergence Update solution values with converged values at current time Requested time steps completed? Take a time step

**FVM - Linear Equation Solvers** • Consider the system of algebraic equations for variable f • The above system of equations is arranged in a matrix and solved iteratively. • For a structured grid, the coefficient matrix is banded. Special line-by-line iterative techniques such as the Line Gauss-Seidel (LGS) method may be used. • LGS method involves solving the equations in a “line” simultaneously. • The equations are set-up in a tri-diagonal matrix solved via Gaussian elimination • For an unstructured grid, no line structure exists. Point-iterative methods are used, e.g., the Point Gauss-Seidel (PGS) technique. • LGS/PGS locally reduce errors but can miss long-wavelength errors. Multigrid acceleration will speed up the LGS/PGS convergence.

**Marching direction** sweeping direction Flow Values from previous sweep Line to be solved Values from previous iteration FVM - Line Gauss-Seidel (LGS) Method • The LGS method is used on structured grids and involves the following steps: • simultaneously solve the equations in the sweep direction • march to next row or column

**FVM - The Multigrid Solver** • The LGS and PGS solvers both transmit the influence of near-neighbors effectively and are less effective at transmitting the influence of far away grid points and boundaries, thereby, slowing convergence. • “Multigrid” solver accelerates convergence for: • Large number of cells • Large cell aspect ratios • x/y > 20 • Large differences in thermal conductivity • Such as in conjugate heat transfer • General concept of multigrid is the same for structured and unstructured grids, although the implementation is different.

**coarse grid level 1** coarse grid level 2 original grid The Multigrid Concept (1) • Multigrid solver uses a sequence of grids going from fine to coarse. • Influence of boundaries and far-away points more easily transmitted to interior on coarse meshes than on fine meshes. • In coarse meshes, grid points are closer together in the computational space and have fewer computational cells between any two spatial locations. • Fine meshes give more accurate solutions.

**corrections** coarse mesh fine mesh summed equations (or volume-averaged solution) The Multigrid Concept (2) • The solutions on the coarser meshes is used as a starting point for solutions on the finer meshes. • Coarse-mesh solution contains influence of boundaries and far neighbors. • These effects felt more easily on coarse mesh. • Accelerates convergence on fine mesh. • Final solution obtained for original (fine) mesh. • Coarse mesh calculations: • only accelerates convergence • do not change final answer

**FVM - Under-relaxation** • Equation set being solved is non-linear. • Equation for one variable may depend on other variables, e.g., • Temperature • Mass fraction • For stability the change in a variable fp value from iteration to iteration is reduced by an “under-relaxation” factor, : • For example, an under-relaxation of 0.2 restricts the change in P to 20% of the computed change of for one iteration.

**FVM - Residuals and Convergence** • At convergence: • All discrete conservation equations (momentum, energy, etc.) are obeyed in all cells to a specified tolerance. • The solution no longer changes with additional iterations. • Mass, momentum, energy and scalar balances are obtained. • “Residuals” measure imbalance (or error) in conservation equations. • Residual at point P is defined as: • An overall measure of the residual in the domain is: • Residuals can be scaled relative to the starting residual

**Finite Element Solution Methods** • We seek a solution to the equation of the form: K(u) u = F • A solution method is made up of two parts • Algorithm: solution organization scheme • Equation solver: solves linear system of equations • We shall consider two algorithms and two equation solvers

**FEM Algorithms and Equation Solvers** • Algorithms: • fully-coupled • segregated • Equation solvers: • Gaussian elimination • Iterative methods: • non-symmetric equation systems • symmetric equation systems (pressure eqns.)

**Fully-Coupled Algorithm (1)** • The most common solution scheme is the so-called Newton-Raphson iteration, or Newton’s method for short • First, re-write the equation as: R(u) = K(u) u - F • Using a Taylor series expansion and some further manipulations, we arrive at:

**Fully-Coupled Algorithm (2)** • Advantages: • converges very rapidly • Disadvantages: • requires good initial guess • calculation of J-1(ui) is expensive • Alternatives: • Modified Newton-Raphson: evaluate J-1(ui) only once • Quasi-Newton: update J-1(ui) in a simple manner graphic representation of Newton’s method

**Segregated Algorithm (1)** • K(u) u = F is never formed • Rather, it is decomposed into a set of decoupled equations: • Kuu - Cxp = fuu momentum equation • Kvv - Cyp = fv v momentum equation • CxTu + CyTv = 0 continuity equation • KTT = fT energy (scalar) equation • No explicit equation for pressure! • Replace continuity equation with Poisson-type pressure matrix equation (derived from manipulating discretized momentum and continuity eqn’s) • The pressure can be calculation in a number of ways

**Segregated Algorithm (2) ** • Pressure projection method • given the current values of u, v and T, obtain an approximate pressure bo solving a discrete pressure equation • relax the pressure, i.e.: • using pnew, solve the momentum equations and energy equation • using the newly computed velocities, solve for the pressure correction, Dp • adjust the velocity field (so that it obeys the incompressibility constraint) using Dp • Advantage: less memory use • Disadvantage: more iterations Each equation set can be solved iteratively (inner iteration) or simulaneously (Gaussian elimination) uvpT outer iteration

**Equation Solvers** • Iterative • Non-symmetric equation systems: • Conjugate gradient squared • GMRES • Symmetric equation systems (pressure): • Conjugate gradient • Conjugate residual • Gaussian elimination

**Underrelaxation** • Two forms are used • explicit (similar to FVM approach) • carries some “history” forward • used with fully-coupled method • also used for pressure in segregated method • implicit • alters the weighting term for matrix diagonal • used for other equations (not pressure) with segregated method

**Convergence** • Various quantities can be used to judge convergence of an FEM solution • The more commonly used are: • Relative change in solution between iterations ||Ui - Ui-1|| / ||Ui|| < tolerance • Relative numerical accuracy (R is residual vector) ||Ri|| / ||R0|| < tolerance