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California Institute of Technology Graduate Aerospace Laboratories California Institute of Technology Kochmann Research Group EAS Division

Ae/AM/CE/ME 214 (Fall/Winter 2016/2017)

Ae/AM/CE/ME 214ab. 9 units (3-0-6); first, second terms. Introduction to the use of numerical methods in the solution of solid mechanics and multiscale mechanics problems. First term covers linear and function spaces. Variational principles. Finite element analysis. Variational problems in linear and finite kinematics. Time integration, initial boundary value problems. Elasticity and inelasticity. Constitutive modeling. Error estimation. Accuracy, stability and convergence. Iterative solution methods. Adaptive strategies. Second term emphasizes multiscale modeling strategies, including computational homogenization in linearized and finite kinematics, spectral methods, atomistics, and atomistic-to-continuum coupling techniques.

contents of Ae214b: Review of computational mechanics. Multiscale modeling strategies, micro-to-macro transition and homogenization problem. Computational homogenization in linear and finite kinematics, thermal problem. Spectral methods. Atomistic modeling and atomistic-to-continuum coupling techniques with applications.

instructor: Prof. Dennis Kochmann (372 Firestone, office hours: anytime)
teaching assistants: Vidyasagar, Greg Phlipot, Carlos Portela
class: Tuesdays & Thursdays, 2:30 - 3:55 pm, 101 Guggenheim
office hours: Wednesdays, 6:00-7:30 pm, 328 SFL
code: We will use the FE code developed in Ae214a.
projects: Course projects will be a major component of this class (to be discussed in class).

course downloads and links: homework problems (from Ae214a/Ae214b):
lecture notes:



Ae102a: Mechanics of Solids and Structures,
Ae160ab: Continuum Mechanics of Fluids and Solids (FA2015/WI2016)

Ae-AM-ME-CE 102a - Ae/AM/CE/Ge/ME 160a. 9 units (3-0-6); first terms. An introduction to continuum mechanics of fluids and solids and the mechanics of solids and structures with engineering applications. First term covers the general kinematics of deformation, gives an overview of the balance laws and of constitutive theory. Second term emphasizes applications (boundary value problems and stability analysis) and discusses specific types of constitutive models (elasticity, linearized elasticity, viscoelasticity, plasticity and thermoelasticity) with a brief introduction to computational mechanics.

contents of 102a/Ae160a: Review of vector and tensor algebra and calculus. Configurations and motion of a body. Kinematics: study of deformations, rotations and stretches, spectral and polar decomposition. Linearized kinematics. Lagrangian and Eulerian strain velocity and spin tensor fields. Kinetics: balance laws, mass conservation, linear and angular momentum, force, traction, notions of stress, equations of motion, equilibrium equations, power theorem. Thermodynamics of bodies: internal energy, heat flux, first law of thermodynamics. Linear theory and linear elasticity.

contents of Ae160b: Constitutive theory in linearized and finite deformations: Coleman-Noll theory, material frame indifference, thermodynamic potentials. Examples of constitutive relations: thermoelastic material, ideal fluids, rigid conductor; elasticity (finite elasticity, symmetry, isotropy, Neo-Hookean, Mooney-Rivlin, St.-Venant), composites, internal constraints and incompressibility, variational forms, boundary value problems; linearized elasticity with internal constraints; Navier-Stokes equations. Elastic wave propagation and elastic stability theory. Viscoelasticity and (visco)plasticity. Introduction to computational mechanics.




Ae108ab: Computational Mechanics (last offered FA2014/WI2015)

Ae/CE/AM 108ab. 9 units (3-0-6); first, second terms Numerical methods and techniques for solving initial boundary value problems in continuum mechanics (from heat conduction to statics and dynamics of solids and structures). Finite difference methods, direct methods, variational methods, finite elements in small strains and at finite deformation for applications in structural mechanics and solid mechanics. Solution of the partial differential equations of heat transfer, solid and structural mechanics, and fluid mechanics. Transient and nonlinear problems. Computational aspects and development and use of finite element code.


course project examples (from WI2015):

HHT implicit dynamics solver, FIRE solver, meshless interpolation, generalized element structure, multi-node constraints, mesh adaptation, corotational beam elements, large-deformation rubbery material models and elements, piezoelectric coupling, finite-deformation electro-mechanical coupling, von Mises plasticity, crystal plasticity, homogenization using affine and periodic boundary conditions.


Figure: Example simulation results from the final projects of the Ae108b class of 2014/15.

example of a simulation using our computational code from previous courses:

A rubbery ball bouncing on a rigid table (Neo-Hookean nonlinear elastic sphere on an elastic foundation, simplicial tetrahedral elements, implicit time integration):