Finite Element Analysis: Concepts and Applications
Iso-parametric, Sub-parametric, and Super-parametric Elements
Iso-parametric Elements
In iso-parametric elements, the geometry and field variables (such as displacement and temperature) are interpolated using the same shape functions. The number of nodes used for defining the geometry is the same as that used for interpolating the field variables.
Example: A 4-node quadrilateral element where both geometry and field variables are interpolated with the same bilinear shape functions.
Sub-parametric Elements
In sub-parametric elements, the geometry is interpolated using fewer nodes compared to the interpolation of the field variables. This is typically used when the geometry is simpler, but the field variables require higher precision.
Example: Using a 4-node element to define the geometry but 8 shape functions for the field variable.
Super-parametric Elements
In super-parametric elements, the geometry is interpolated using more nodes compared to the field variables. This is useful when representing complex geometries while keeping the field interpolation simpler.
Example: Using an 8-node element for geometry interpolation but only 4 shape functions for the field variable.
Shape Functions in Finite Element Analysis
Shape functions are mathematical functions used to interpolate the field variables (like displacement or temperature) within a finite element. They are expressed in terms of the element’s local coordinates (ξ, η, ζ).
Properties of Shape Functions
- Partition of Unity: The sum of all shape functions for an element equals
- Non-Negativity: Shape functions are always non-negative within the element domain.
- Continuity: Shape functions must be continuous within the element and across its boundaries to ensure compatibility.
- Interpolation: The field variable is interpolated as
Accuracy Improvement Methods: h-Method and p-Method
h-Method
The h-method improves accuracy by increasing the number of elements, thereby reducing the element size (h). The order of the polynomial interpolation remains the same. This method is ideal for problems with geometric complexity.
p-Method
The p-method improves accuracy by increasing the order of the shape functions (polynomial degree p). The number of elements remains constant. This method is effective for problems with smooth solutions.
Comparison of h-Method and p-Method
- h-method: Better for capturing local features or discontinuities.
- p-method: Efficient for smooth problems, requiring fewer degrees of freedom.
Jacobian Matrix in FEM
Significance of the Jacobian Matrix
- Coordinate Transformation: Converts natural coordinates (ξ, η) into global coordinates (x, y).
- Element Geometry: Ensures the correct mapping of curved or irregular elements in global space.
- Integration: Facilitates numerical integration in natural coordinates by transforming integration bounds.
- Strain-Displacement Relations: Relates strains in natural coordinates to strains in global coordinates.
General Procedure of the Finite Element Method
- Discretization of the Domain: Divide the domain into finite elements (e.g., triangles, quadrilaterals). Choose appropriate element types.
- Selection of Shape Functions: Define shape functions to approximate field variables within each element.
- Derivation of Element Equations: Use methods like Galerkin or energy principles to formulate equations for each element.
- Assembly of Global Equations: Combine element equations into a global system and enforce continuity.
- Application of Boundary Conditions: Apply constraints (e.g., displacements, forces) to modify the global system.
- Solution of the Global System: Solve [K]{u}={F} for unknown nodal values.
- Post-Processing: Calculate secondary quantities (e.g., stresses, strains) and visualize results.
Sources of Error in the Finite Element Method
- Discretization Error: Due to the approximation of the domain using finite elements. Depends on element size (h) and order of interpolation (p).
- Modeling Error: Arises from simplifications in material properties, boundary conditions, or loading.
- Numerical Integration Error: Occurs when using approximate integration techniques like Gaussian quadrature.
- Round-off Error: Results from finite precision arithmetic in computer calculations.
- Convergence Issues: Errors due to improper meshing, leading to non-convergence in solutions.
Principle of Minimum Total Potential Energy
Statement
For a deformable body in equilibrium, the total potential energy (Π) is minimized:
Π = U – W
where U is the strain energy stored in the body, and W is the work done by external forces.
Application in FEM
- Discretize the domain into finite elements.
- Approximate the displacement field using shape functions.
- Formulate the total potential energy (Π) for the system.
- Derive equilibrium equations by minimizing Π with respect to nodal displacements.
Calculating Linear Interpolation Functions
Boundary Conditions in FEM
Boundary Conditions are constraints applied to a problem to specify the behavior of a system at its boundaries. These are necessary to solve a differential equation uniquely.
Types of Boundary Conditions
- Essential Boundary Conditions (Dirichlet Conditions): Specify the value of a variable (e.g., displacement in structural analysis, temperature in heat transfer). Example: u=0 at a fixed boundary.
- Natural Boundary Conditions (Neumann Conditions): Specify the value of a derivative or flux (e.g., stress, heat flux). Example: ∂u/∂x = q
- Mixed Boundary Conditions: Combination of Dirichlet and Neumann conditions. Example: A boundary with both fixed displacement and applied stress.
Mass Matrices: Consistent vs. Lumped
Consistent Mass Matrix
- Derived by applying the same shape functions used for stiffness matrix derivation.
- Distributes mass proportionally across all nodes of the element.
- Results in higher accuracy for dynamic problems.
- Example (for a 1D element):
Lumped Mass Matrix
- Simplifies the mass matrix by assigning the total mass of the element to its nodes (diagonal matrix).
- Easier to implement and computationally efficient but less accurate.
- Example (for a 1D element):
Comparison
- Consistent Mass Matrix: Accurate but computationally expensive.
- Lumped Mass Matrix: Less accurate but computationally efficient.
Deriving Shape Functions for an Eight-Node Element
An eight-node rectangular finite element includes:
- Four corner nodes (1, 2, 3, 4).
- Four mid-side nodes (5, 6, 7, 8).
The natural coordinates (ξ, η) are defined as follows:
- Corner nodes:
- Node 1: (-1, -1)
- Node 2: (1, -1)
- Node 3: (1, 1)
- Node 4: (-1, 1)
- Mid-side nodes:
- Node 5: (0, -1)
- Node 6: (1, 0)
- Node 7: (0, 1)
- Node 8: (-1, 0)
General Structure of Shape Functions
Corner Nodes (Nodes 1 to 4): 
Mid-Side Nodes (Nodes 5 to 8): 
Shape Function Derivation
Corner Nodes: 
Mid-Side Nodes: 
Why FEM is an Approximate Solution
FEM is an approximate solution method because:
- Discretization of the Domain: The continuous domain is divided into a finite number of elements, introducing discretization errors.
- Use of Shape Functions: Field variables within each element are approximated using shape functions, which may not capture the exact variation.
- Numerical Integration: Approximation is introduced when evaluating integrals over elements.
Improving Accuracy in FEM
- Mesh Refinement (h-method): Use finer meshes by increasing the number of smaller elements.
- Higher-Order Elements (p-method): Use higher-order shape functions for better approximation.
- Adaptive Meshing: Refine the mesh adaptively in regions with higher gradients or errors.
- Accurate Boundary Conditions: Ensure boundary conditions are implemented correctly.
- Error Estimation: Use error estimation techniques to guide refinement.
Convergence in Finite Element Analysis
Convergence in FEA refers to the process of approaching the exact solution as the approximation is improved.
- h-Convergence: Achieved by refining the mesh (reducing element size). Smaller elements provide better resolution of the domain.
- p-Convergence: Achieved by increasing the order of the polynomial shape functions. Suitable for problems with smooth solutions.
- hp-Convergence: Combines both h- and p-convergence for rapid improvement. Refines the mesh and uses higher-order elements.
Deriving Shape Functions for a Four-Node Element
The natural coordinate system is defined on a unit square, where:
ξ, η ∈ [-1, 1]
The four corners of the element are located at:
- Node 1: (ξ, η) = (-1, -1)
- Node 2: (ξ, η) = (1, -1)
- Node 3: (ξ, η) = (1, 1)
- Node 4: (ξ, η) = (-1, 1)
General Form for a Bilinear Four-Node Element
Deriving Shape Functions for Each Node
