Finite Element Method (FEM)

6.

Application in Heat Transfer and Other Field Problems

6.1.

Heat Transfer Fundamentals

6.1.1.

Heat Conduction Mechanisms

6.1.2.

6.1.3.

Heat Equation Derivation

6.1.4.

Thermal Properties

6.2.

Steady-State Heat Conduction

6.2.1.

Governing Equations

6.2.2.

Boundary Conditions

6.2.2.1.

Prescribed Temperature

6.2.2.2.

Prescribed Heat Flux

6.2.2.3.

Convection Boundaries

6.2.2.4.

Radiation Boundaries

6.2.3.

FEM Formulation

6.2.3.1.

Element Conductivity Matrix

6.2.3.2.

Element Heat Generation Vector

6.2.3.3.

Assembly Process

6.3.

Transient Heat Conduction

6.3.1.

Time-Dependent Heat Equation

6.3.2.

Initial Conditions

6.3.3.

FEM Spatial Discretization

6.3.4.

Time Integration Schemes

6.3.4.1.

Forward Euler Method

6.3.4.2.

Backward Euler Method

6.3.4.3.

Crank-Nicolson Method

6.3.4.4.

6.3.5.

Stability and Accuracy Analysis

6.3.6.

Mass Matrix Formulation

6.3.6.1.

Consistent Mass Matrix

6.3.6.2.

Lumped Mass Matrix

6.4.

Convection Heat Transfer

6.4.1.

Convection Boundary Conditions

6.4.2.

Heat Transfer Coefficients

6.4.3.

Implementation in FEM

6.5.

Radiation Heat Transfer

6.5.1.

Stefan-Boltzmann Law

6.5.2.

6.5.3.

Nonlinear Radiation Boundaries

6.5.4.

Linearization Techniques

6.6.

Other Field Applications

6.6.1.

Seepage and Groundwater Flow

6.6.1.1.

6.6.1.2.

Governing Equations

6.6.1.3.

Hydraulic Conductivity

6.6.1.4.

FEM Implementation

6.6.2.

Torsion of Shafts

6.6.2.1.

Prandtl Stress Function

6.6.2.2.

Governing Equations

6.6.2.3.

Boundary Conditions

6.6.2.4.

FEM Formulation

6.6.3.

Potential Flow Problems

6.6.3.1.

Laplace Equation

6.6.3.2.

Velocity Potential

6.6.3.3.

Stream Function

6.6.4.

Acoustic Problems

6.6.4.1.

6.6.4.2.

Helmholtz Equation

6.6.4.3.

Acoustic Boundary Conditions

6.6.4.4.

FEM Implementation

6.6.5.

Electrostatic Problems

6.6.5.1.

Poisson's Equation

6.6.5.2.

Electric Potential

6.6.5.3.

Dielectric Materials

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7. Advanced Topics in FEM