6.3 Finite element method

The finite element method is a powerful numerical technique for solving complex partial differential equations in multiphase flow modeling. It discretizes the domain into smaller elements, approximating solutions using interpolation functions. This approach offers flexibility in handling intricate geometries and boundary conditions.

FEM's versatility stems from its ability to adapt to various problem types through mesh generation, element selection, and interpolation functions. It excels in modeling multiphase flows by discretizing equations for each phase, coupling them, and capturing interfacial phenomena. Advanced techniques like adaptive mesh refinement further enhance its capabilities.

Basics of finite element method

• Numerical method for solving partial differential equations (PDEs) by discretizing the domain into smaller elements
• Approximates the solution using a finite number of elements with simple geometry and interpolation functions
• Widely used in various fields of engineering, including multiphase flow modeling, due to its flexibility and ability to handle complex geometries and boundary conditions

Discretization of domain

Mesh generation techniques

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• Divides the computational domain into smaller, simpler elements (triangles, quadrilaterals, tetrahedra, hexahedra)
• Structured meshes have regular connectivity and are easier to generate but less flexible in adapting to complex geometries
• Unstructured meshes have irregular connectivity and are more adaptable to complex geometries but require more sophisticated mesh generation algorithms
• Hybrid meshes combine structured and unstructured elements to balance the advantages of both approaches

Types of finite elements

• Linear elements (1D: lines, 2D: triangles and quadrilaterals, 3D: tetrahedra and hexahedra) are the simplest and most commonly used
• Higher-order elements (quadratic, cubic) provide better accuracy but increase computational cost
• Special elements (infinite elements, shell elements) are used for specific applications or to handle unique geometric features

Node and element numbering

• Nodes are the vertices of the elements and store the primary variables (displacements, pressures, temperatures)
• Elements are numbered to establish connectivity between nodes and facilitate the assembly of global equations
• Proper node and element numbering schemes can improve computational efficiency and minimize bandwidth of the global matrices

Interpolation functions

Linear interpolation functions

• Simplest and most widely used interpolation functions
• Assume a linear variation of the primary variables within the element
• Require only the nodal values to determine the value at any point within the element
• Computationally efficient but may require finer meshes for accurate results

Higher-order interpolation functions

• Assume a higher-order (quadratic, cubic) variation of the primary variables within the element
• Require additional nodes within the element (mid-side nodes for quadratic elements) to determine the value at any point
• Provide better accuracy and can capture curved geometries more effectively
• Increase the computational cost and complexity of the formulation

Formulation of element equations

Weak form of governing equations

• Transforms the strong form (PDEs) into an integral form by multiplying with a weight function and integrating over the domain
• Relaxes the continuity requirements on the solution and allows for a wider range of approximation functions
• Forms the basis for the finite element formulation and the development of element equations

Galerkin method

• A specific choice of weight functions, where the weight functions are chosen to be the same as the interpolation functions
• Leads to a symmetric and positive-definite global stiffness matrix, which is advantageous for numerical solution
• Widely used in finite element formulations due to its simplicity and good convergence properties

Boundary conditions implementation

• Essential (Dirichlet) boundary conditions are imposed by modifying the global equations, typically by eliminating the corresponding degrees of freedom
• Natural (Neumann) boundary conditions are incorporated into the weak form and contribute to the global load vector
• Mixed boundary conditions (Robin) can be handled by a combination of essential and natural boundary condition techniques

Assembly of global equations

Element connectivity

• Determines how the local element equations are combined to form the global system of equations
• Defined by the node and element numbering scheme and the element topology
• Crucial for the efficient assembly of the global matrices and vectors

Global stiffness matrix

• Obtained by summing the contributions of the local element stiffness matrices according to the element connectivity
• Symmetric and sparse matrix that represents the discrete form of the governing equations
• Size depends on the number of nodes and the degrees of freedom per node

Global load vector

• Obtained by summing the contributions of the local element load vectors according to the element connectivity
• Incorporates the effect of external loads, source terms, and natural boundary conditions
• Size depends on the number of nodes and the degrees of freedom per node

Solution of global equations

Direct methods

• Solve the global system of equations by factorizing the global stiffness matrix (LU decomposition, Cholesky decomposition)
• Provide an exact solution (up to machine precision) but can be computationally expensive for large systems
• Suitable for small to medium-sized problems or when high accuracy is required

Iterative methods

• Solve the global system of equations by iteratively improving an initial guess until convergence is achieved
• Require less memory and computational effort compared to direct methods, especially for large and sparse systems
• Examples include Jacobi, Gauss-Seidel, and Krylov subspace methods (Conjugate Gradient, GMRES)

Convergence criteria

• Determine when the iterative solution process has reached a satisfactory level of accuracy
• Can be based on the residual norm, relative change in the solution, or a combination of both
• Choice of convergence criteria affects the accuracy and computational cost of the solution process

Post-processing of results

Interpolation of nodal values

• Computes the values of the primary variables at any point within the elements using the interpolation functions and the nodal values
• Allows for the visualization and analysis of the solution field within the computational domain
• Can be used to extract values at specific locations or along lines/surfaces of interest

Calculation of derived quantities

• Computes secondary quantities (stresses, strains, velocities, fluxes) from the primary variables using the appropriate constitutive relations or gradient operators
• Provides additional insight into the behavior of the system and allows for the assessment of design criteria or performance metrics
• May require the use of special post-processing techniques (stress recovery, superconvergent patch recovery) to improve the accuracy of the derived quantities

Visualization techniques

• Graphical representation of the solution field and derived quantities to facilitate the interpretation and communication of the results
• Includes contour plots, surface plots, vector plots, and streamlines for multiphase flow applications
• Can be enhanced with animation, interactive tools, and virtual reality techniques to provide a more immersive and intuitive understanding of the results

Advantages vs disadvantages

Flexibility in geometry and boundary conditions

• Can handle complex geometries and irregular boundaries by using unstructured or adaptive meshes
• Allows for the application of various types of boundary conditions (Dirichlet, Neumann, Robin) on different parts of the domain
• Facilitates the modeling of multi-physics problems by accommodating different governing equations and coupling conditions

Ability to handle complex materials

• Can incorporate various constitutive models for linear and nonlinear materials (elasticity, plasticity, viscoelasticity)
• Allows for the modeling of heterogeneous materials with varying properties by assigning different material parameters to different elements
• Enables the simulation of multi-component systems (composites, porous media) by using special elements or homogenization techniques

Computational cost considerations

• Requires the solution of large systems of equations, which can be computationally expensive, especially for 3D problems or high-resolution meshes
• The computational cost increases with the number of elements, the order of the interpolation functions, and the complexity of the governing equations
• Can be mitigated by using efficient solution algorithms, parallel computing, or adaptive mesh refinement techniques

Applications in multiphase flow

Discretization of multiphase equations

• Involves the formulation of the governing equations for each phase (mass, momentum, energy) and the coupling terms between phases
• Requires the use of appropriate interpolation functions and numerical schemes to handle the different scales and physics of the phases
• May involve the use of special elements (interface elements) or enrichment techniques (extended finite element method) to capture the interface between phases

Coupling of phases

• Ensures the continuity of primary variables (velocity, pressure, temperature) and fluxes (mass, momentum, energy) across the interface between phases
• Can be achieved by using interface conditions, penalty methods, or Lagrange multipliers
• Requires the consistent discretization and assembly of the coupling terms in the global equations

Modeling of interfacial phenomena

• Includes surface tension, phase change, and mass/heat transfer between phases
• Requires the accurate representation of the interface geometry and the incorporation of the appropriate jump conditions in the finite element formulation
• May involve the use of level set, volume of fluid, or phase field methods to track the evolution of the interface

Advanced topics

Adaptive mesh refinement

• Dynamically adjusts the mesh resolution based on error indicators or physical criteria to improve the accuracy and efficiency of the solution
• Involves the refinement (h-adaptivity) or coarsening of the mesh, the redistribution of nodes (r-adaptivity), or the adjustment of the interpolation order (p-adaptivity)
• Particularly useful for multiphase flow problems with localized features (interfaces, boundary layers, singularities) that require high resolution

Parallel computing in FEM

• Exploits the inherent parallelism in the finite element method by distributing the computational tasks among multiple processors or cores
• Involves the partitioning of the mesh, the distribution of the global matrices and vectors, and the parallel solution of the global equations
• Enables the simulation of large-scale multiphase flow problems that are intractable on a single processor

Stabilization techniques for multiphase flows

• Address the numerical instabilities that arise from the discretization of the multiphase equations, especially in the presence of strong convection or sharp interfaces
• Include upwind schemes, streamline-upwind Petrov-Galerkin (SUPG) methods, and variational multiscale (VMS) methods
• Ensure the stability and accuracy of the finite element solution for a wide range of flow conditions and physical parameters