八月 6

Unit 1: Linear, elliptic partial differential equations in one dimension. Elasticity, heat conduction and mass diffusion

Linear Elliptic Partial Differential Equations in one dimension.

Examples:

  1. 1D heat conduction at steady state.
  2. 1D mass diffusion at steady state.
  3. 1D elasticity at steady state.

Dirichlet Boundary Condition(Displacement Boundary Condition): applied to primal field.

Neumann Boundary Condition(Traction Boundary Condition): applied to the derivative of the primal field.

We do not have situations where we have Neumann Boundary Conditions at both ends, because of rigid body motion. Neumann Boundary Conditions can be specified alone in time-dependent elasticity problems.

Strong From: 1. The PDE has to hold at every point in the interval of interest. 2. The primal field has to hold “strong condition” of smoothness, able to take two derivatives.

Strong Form

Weak Form: Basis of the Finite Element Methods.

Weak Form