8.3 Finite element methods

Finite element methods are powerful numerical techniques for solving complex fluid dynamics problems. By discretizing the domain into smaller elements, FEM approximates partial differential equations governing fluid flow, enabling analysis of intricate geometries and boundary conditions.

FEM involves key steps: discretizing the domain, formulating governing equations in weak form, and assembling a global system of equations. This approach allows for accurate solutions to challenging fluid dynamics problems, from incompressible flows to turbulence modeling and fluid-structure interaction.

Overview of finite element methods

• Finite element methods (FEM) are numerical techniques used to solve complex engineering problems, including fluid dynamics, by discretizing the domain into smaller, simpler elements
• FEM allows for the approximation of partial differential equations (PDEs) governing fluid flow, heat transfer, and structural mechanics, enabling the analysis of complex geometries and boundary conditions
• The method involves formulating the governing equations, discretizing the domain, assembling the global system of equations, and solving the resulting linear or nonlinear system

Fundamental concepts in FEM

Discretization of domain into elements

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• The computational domain is divided into a finite number of smaller, simpler subdomains called elements (triangles, quadrilaterals, tetrahedra, or hexahedra)
• Elements are connected at nodes, which are points where the unknown variables (velocity, pressure, temperature) are calculated
• Discretization allows for the approximation of the continuous problem using a finite number of degrees of freedom

Shape functions for element interpolation

• Shape functions are used to interpolate the unknown variables within each element based on the nodal values
• Linear, quadratic, or higher-order shape functions can be employed, depending on the desired accuracy and computational cost
• Shape functions ensure continuity of the solution across element boundaries and enable the mapping between local and global coordinate systems

Local and global coordinate systems

• Local coordinate systems are defined for each element, simplifying the integration and evaluation of shape functions
• Global coordinate system represents the entire computational domain and is used for assembling the global system of equations
• Coordinate transformations are performed to map between local and global systems, ensuring compatibility of the solution across elements

Formulation of governing equations

Weak form of partial differential equations

• The strong form of the governing PDEs is transformed into a weak form by multiplying the equations by a test function and integrating over the domain
• The weak form relaxes the continuity requirements on the solution, allowing for the use of simpler, piecewise-continuous approximations
• Boundary conditions are naturally incorporated into the weak form through the boundary integrals

Galerkin method of weighted residuals

• The Galerkin method is a specific choice of test functions, where the test functions are chosen to be the same as the shape functions used for interpolation
• This approach leads to a symmetric, positive-definite system of equations for many problems, which is advantageous for numerical solution
• The Galerkin method minimizes the residual of the governing equations in a weighted sense, ensuring optimal approximation of the solution

Boundary conditions and constraints

• Essential (Dirichlet) boundary conditions prescribe the values of the unknown variables at specific nodes or boundaries
• Natural (Neumann) boundary conditions specify the fluxes or gradients of the unknown variables at the boundaries
• Constraints, such as incompressibility or no-slip conditions, are imposed using techniques like the penalty method or Lagrange multipliers
• Proper treatment of boundary conditions and constraints is crucial for obtaining accurate and physically meaningful solutions

Element types and characteristics

1D, 2D, and 3D elements

• 1D elements (lines) are used for problems with one dominant spatial dimension, such as beams or trusses
• 2D elements (triangles or quadrilaterals) are employed for planar or axisymmetric problems, like heat conduction or elasticity
• 3D elements (tetrahedra or hexahedra) are utilized for complex, three-dimensional geometries encountered in fluid dynamics or structural analysis

Linear vs higher-order elements

• Linear elements have nodes only at the vertices and provide a piecewise-linear approximation of the solution
• Higher-order elements (quadratic, cubic) include additional nodes along the edges or faces, enabling a more accurate representation of the solution
• The choice between linear and higher-order elements depends on the required accuracy, computational cost, and the smoothness of the expected solution

Isoparametric elements and mapping

• Isoparametric elements employ the same shape functions for both geometry and solution interpolation
• This approach allows for the representation of curved boundaries and distorted elements using a single mapping from the reference element to the physical element
• Isoparametric mapping simplifies the integration and evaluation of element matrices, as it is performed on a standard reference element

Assembly of global system

Element connectivity and numbering

• Element connectivity defines how the elements are connected to each other through shared nodes
• A global numbering system is established for nodes and elements, ensuring compatibility of the solution across the domain
• Connectivity arrays store the relationship between local (element-level) and global (domain-level) degrees of freedom

Global stiffness matrix and load vector

• The global stiffness matrix is assembled by summing the contributions from each element's local stiffness matrix
• The global load vector is constructed by aggregating the element-level load vectors, which include body forces, surface tractions, and boundary conditions
• Assembly process ensures continuity of the solution and equilibrium of forces at the nodes

Sparse matrix storage techniques

• The global stiffness matrix is typically sparse, with many zero entries, due to the local nature of element interactions
• Sparse matrix storage techniques, such as compressed row storage (CRS) or compressed column storage (CCS), are employed to efficiently store and manipulate the global matrix
• These techniques reduce memory requirements and computational costs associated with solving large systems of equations

Solution techniques for linear systems

Direct vs iterative solvers

• Direct solvers, such as Gaussian elimination or LU decomposition, compute the exact solution of the linear system in a finite number of steps
• Iterative solvers, like the conjugate gradient method or Gauss-Seidel method, approximate the solution through successive iterations until a desired level of accuracy is achieved
• The choice between direct and iterative solvers depends on the size and sparsity of the system, available computational resources, and required accuracy

Gaussian elimination and LU decomposition

• Gaussian elimination is a classic direct solver that systematically eliminates variables to obtain an upper triangular system, which is then solved by back-substitution
• LU decomposition factorizes the matrix into a product of a lower triangular matrix (L) and an upper triangular matrix (U), enabling efficient solution of the system
• These methods are robust and accurate but can be computationally expensive for large systems

Conjugate gradient method and preconditioning

• The conjugate gradient (CG) method is an iterative solver well-suited for symmetric, positive-definite systems arising from FEM discretizations
• CG minimizes the energy norm of the error and exhibits rapid convergence for well-conditioned systems
• Preconditioning techniques, such as incomplete LU factorization or multigrid methods, are used to improve the conditioning of the system and accelerate convergence

Adaptive mesh refinement strategies

Error estimation and indicators

• Error estimation techniques, such as a posteriori error estimators or residual-based indicators, are used to assess the quality of the FEM solution
• These methods quantify the local error in each element based on the residual of the governing equations or the jump in the solution across element boundaries
• Error indicators guide the adaptive refinement process by identifying regions where the mesh needs to be refined to improve accuracy

h-refinement vs p-refinement

• h-refinement involves subdividing elements into smaller ones in regions with high error, increasing the spatial resolution of the mesh
• p-refinement increases the polynomial order of the shape functions within elements, improving the approximation accuracy without changing the mesh topology
• hp-refinement combines both strategies, adaptively adjusting the mesh size and polynomial order to optimize the trade-off between accuracy and computational cost

Automatic mesh generation and optimization

• Automatic mesh generation algorithms, such as Delaunay triangulation or advancing front methods, create high-quality meshes based on the geometry and desired element size
• Mesh optimization techniques, like smoothing or topological modifications, improve the quality of the mesh by adjusting node positions or element connectivities
• These methods ensure that the mesh is suitable for FEM analysis, with well-shaped elements and appropriate resolution in critical regions

Stabilization methods for convection-dominated flows

Upwinding and Petrov-Galerkin formulations

• Upwinding techniques modify the weighting functions in the Galerkin formulation to account for the direction of flow and suppress numerical oscillations
• Petrov-Galerkin methods employ different trial and test functions, with the test functions designed to add numerical dissipation in the streamline direction
• These approaches improve the stability and accuracy of FEM for convection-dominated problems, such as high-speed flows or transport phenomena

Streamline-Upwind Petrov-Galerkin (SUPG) method

• SUPG is a popular stabilization method that adds a streamline-dependent perturbation to the test functions, introducing numerical diffusion along the flow direction
• The SUPG formulation maintains consistency with the original governing equations and reduces numerical oscillations without excessive smearing of the solution
• Stabilization parameters in SUPG are typically based on the element Peclet number, which quantifies the relative importance of convection and diffusion

Galerkin Least Squares (GLS) method

• GLS is another stabilization technique that adds a least-squares form of the residual to the Galerkin formulation, minimizing the residual in a weighted sense
• The GLS method is consistent, meaning it does not alter the original governing equations, and provides improved stability and accuracy for convection-dominated flows
• GLS can be applied to a wide range of problems, including incompressible and compressible flows, and is compatible with various element types and orders

Verification and validation of FEM results

Convergence analysis and mesh independence

• Convergence analysis assesses the behavior of the FEM solution as the mesh is refined, ensuring that the numerical approximation approaches the true solution
• Mesh independence studies are conducted by systematically refining the mesh and comparing the solutions to verify that the results are not sensitive to further refinement
• Convergence rates can be examined to confirm that the FEM implementation is correct and to estimate the discretization error

Comparison with analytical solutions

• Analytical solutions, when available, provide a benchmark for verifying the accuracy of FEM results
• Simple test cases with known analytical solutions, such as laminar flow in a channel or heat conduction in a rectangular domain, are used to validate the FEM formulation and implementation
• Comparing FEM results with analytical solutions helps identify potential sources of error and builds confidence in the numerical model

Experimental validation and uncertainty quantification

• Experimental validation involves comparing FEM predictions with physical measurements or observations to assess the accuracy and reliability of the numerical model
• Validation experiments are designed to test specific aspects of the FEM model, such as boundary conditions, material properties, or flow regimes
• Uncertainty quantification techniques, like sensitivity analysis or Bayesian inference, are employed to characterize the impact of input uncertainties on the FEM results and establish confidence intervals

Advanced topics in FEM for fluid dynamics

Incompressible Navier-Stokes equations

• The incompressible Navier-Stokes equations govern the motion of viscous, incompressible fluids and are a fundamental model in fluid dynamics
• FEM formulations for the Navier-Stokes equations often employ mixed interpolation, with different shape functions for velocity and pressure to satisfy the LBB (inf-sup) stability condition
• Techniques like the projection method or the SIMPLE algorithm are used to handle the pressure-velocity coupling and enforce the incompressibility constraint

Turbulence modeling and large eddy simulation

• Turbulence modeling is essential for simulating high-Reynolds-number flows, where direct resolution of all scales of motion is computationally infeasible
• Reynolds-Averaged Navier-Stokes (RANS) models, such as k-epsilon or k-omega, introduce additional transport equations for turbulence quantities and provide closure for the averaged equations
• Large Eddy Simulation (LES) directly resolves the large-scale turbulent structures while modeling the effects of smaller scales using subgrid-scale (SGS) models, providing a more accurate representation of turbulence

Fluid-structure interaction and moving meshes

• Fluid-structure interaction (FSI) problems involve the coupled dynamics of fluids and deformable structures, such as blood flow in arteries or wind turbine blades
• FEM formulations for FSI often employ partitioned or monolithic approaches, depending on the strength of the coupling between the fluid and structure domains
• Moving mesh techniques, such as the Arbitrary Lagrangian-Eulerian (ALE) method or the immersed boundary method, are used to handle the deformation of the computational domain and maintain mesh quality