Advanced Finite Element Analysis

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Explore advanced computational mechanics through this comprehensive graduate-level course covering sophisticated finite element analysis techniques for complex engineering problems. Master linear elastic finite element fundamentals including weak form derivations, Galerkin methods, interpolation functions, matrix equations of motion, integral mapping, global stiffness assembly, boundary condition implementation, and stress-strain computations. Delve into nonlinear finite element analysis in one-dimensional systems, learning Newton's method applications, Lagrangian versus Eulerian mesh approaches, total and updated Lagrangian formulations, conservation equations, constitutive laws, and explicit central difference solution methods. Advance to three-dimensional nonlinear analysis by studying continuum kinematics, reference frame transformations, Lagrange finite strain tensors, displacement-based strain formulations, rate of deformation concepts, Cauchy stress formulations, various stress measures, and both total and updated Lagrangian formulations in multi-dimensional contexts. Investigate dynamic finite element analysis including natural frequency calculations, mode shape determination, Newmark-beta implicit integration schemes, damping mechanisms, and mode superposition techniques for modal analysis. Build theoretical foundations through variational methods covering functional extremization, fundamental lemmas of variational calculus, Euler-Lagrange equations, first integrals, delta operators, natural boundary conditions, higher-order derivatives, multiple variable systems, stationary total potential energy principles, elastic body potential energy, Rayleigh-Ritz methods, and connections between Ritz methods and finite element analysis.

Syllabus

Linear Elastic Finite Element Analysis (Overview)
1-1: Linear Elastic Finite Element Analysis (Weak Form of Governing Differential Equation)
1-2: Linear Elastic Finite Element Analysis (Galerkin Method)
1-3: Linear Finite Element Analysis (Interpolation Functions)
1-4a: Linear Finite Element Analysis (Matrix Equations of Motion - Part I)
1-4b: Linear Finite Element Analysis (Matrix Equations of Motion - Part II)
1-5a: Linear Finite Element Analysis (Mapping Integrals - Part I)
1-5b: Linear Finite Element Analysis (Mapping Integrals - Part II)
1-6: Linear Finite Element Analysis (Assembly of Global Stiffness Equations)
1-7: Linear Finite Element Analysis (Applying Boundary Conditions)
1-8: Linear Finite Element Analysis (Computing Stresses and Strains)
2-0: Nonlinear Finite Elements in 1-D (Overview)
2-1: Nonlinear Finite Elements in 1-D (Newton's Method in 1-D)
2-2: Nonlinear Finite Elements in 1-D (Newton's Method for Systems of Equations)
2-3: Nonlinear Finite Elements in 1-D (Lagrangian vs. Eulerian Meshes)
2-4: Nonlinear Finite Elements in 1-D (Total Lagrangian vs. Updated Lagrangian)
2-5a: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - Problem Setup)
2-5b: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - Conservation Equations)
2-5c: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - Constitutive Law)
2-5d: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - Boundary & Initial Cond.)
2-5e: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - Weak Form)
2-5f: Nonlinear Finite Elements in 1-D (Total Lagrangian Formulation - FE Discretization)
2-6: Nonlinear Finite Elements in 1-D (Element and Global Vectors and Matrices)
2-7: Nonlinear Finite Elements in 1-D (Solution Methods - Explicit Central Difference)
2-8a: Nonlinear Finite Elements in 1-D (Updated Lagrangian - Governing Equations)
2-8b: Nonlinear Finite Elements in 1-D (Updated Lagrangian - Weak Form and FE Discretization)
2-8c: Nonlinear Finite Elements in 1-D (Updated Lagrangian - Mesh Distortion)
1-2a: Continuum Kinematics (Reference Frames and Deformation)
1-2b: Continuum Kinematics (Lagrange Finite Strain Tensor)
1-2c: Continuum Kinematics (Meaning of the Lagrange Finite Strain Tensor)
1-2d: Continuum Kinematics (Displacement-Based Strain Formulation)
3-1e: Nonlinear Finite Elements in 3-D (Continuum Kinematics - Rate of Deformation/Velocity Strain)
Continuum Stresses (Cauchy Stress Formula)
3-1g: Nonlinear Finite Elements in 3-D (Continuum Stresses - Stress Measures)
3-1h: Nonlinear Finite Elements in 3-D (Continuum Stresses - Example)
3-2: Nonlinear Finite Elements in 3-D (Total Lagrangian Formulation)
3-3: Nonlinear Finite Elements in 3-D (Updated Lagrangian Formulation)
4-1: Dynamic Finite Element Analysis (Natural Frequencies and Mode Shapes)
4-2: Dynamic FEA (Newmark-beta Implicit Integration)
4-3: Dynamic FEA (Damping)
4-4: Dynamic FEA (Mode Superposition - Modal Analysis)
Variational Methods (Functionals and Extremization)
Variational Methods (Fundamental Lemma of Variational Calculus)
Variational Methods (Example - Shortest Path)
Variational Methods (Example - Surface of Revolution)
Variational Methods (First Integrals of the Euler-Lagrange Equation)
Variational Methods (Delta Operator)
Variational Methods (Natural Boundary Conditions)
Variational Methods (Functionals with Higher Order Derivatives)
Variational Methods (Functionals with Multiple Dependent Variables)
Variational Methods (Functionals with Multiple Independent Variables)
Variational Methods (Principle of Stationary Total Potential Energy)
Variational Methods (Potential Energy of an Elastic Body)
Variational Methods (Rayleigh-Ritz Method)
Variational Methods (Ritz Method and Finite Element Analysis)