FEM Basics Flashcards
(24 cards)
What is the strong form of 1D heat flow?
The strong form of 1D heat flow is a differential equation derived from a heat balance. It is expressed as: -d/dx(k * dT/dx) + q = 0, where k is the thermal conductivity, T is the temperature, and q is the heat generation per unit volume. Boundary conditions include essential (Dirichlet: specifying temperature, e.g., T(0) = T0), natural (Neumann: specifying heat flux, e.g., -kdT/dx = q_flux), and Robin (combined temperature and heat flux, e.g., -kdT/dx + h*(T - T_inf) = 0).
What is the weak form of 1D heat flow?
The weak form is derived by multiplying the strong form by a test function v(x), integrating over the domain, and applying integration by parts to reduce the derivative order. The resulting equation is integral(k * dT/dx * dv/dx) = integral(q * v) + integral(v * q_flux). Essential boundary conditions are applied directly to the solution, while natural boundary conditions contribute to the boundary integrals.
How is the finite element (FE) form of 1D heat flow derived?
The FE form approximates the temperature T(x) using shape functions N_i(x): T(x) = sum(N_i(x) * T_i), where T_i are nodal temperatures. Substituting this into the weak form and using numerical integration leads to a system of algebraic equations: [K]{T} = {F}, where [K] is the stiffness matrix, {T} is the nodal temperature vector, and {F} is the load vector. Boundary conditions are applied by modifying {T} and {F}.
What other 1D problems have similar derivations to heat flow?
Similar problems include: (1) Moisture transport: analogous to heat conduction with moisture concentration replacing temperature. (2) 1D elastic bar: governed by -d/dx(E * du/dx) = f, where E is Young’s modulus, u is displacement, and f is body force. (3) Radial heat transfer: involves a varying cross-sectional area, modifying the heat balance equation.
What is the strong form of stationary and transient 2D heat flow?
The strong form for stationary 2D heat flow is -∇·(k∇T) + q = 0, where k is thermal conductivity, T is temperature, and q is heat generation per unit volume. For transient problems, an additional term accounts for time dependence: ρc * dT/dt - ∇·(k∇T) + q = 0, where ρ is density and c is specific heat.
What is the weak form of stationary and transient 2D heat flow?
The weak form is obtained by multiplying the strong form by a test function v and applying integration by parts. For stationary problems: integral(k * ∇T · ∇v) = integral(q * v) + integral(v * q_flux). For transient problems, an additional term, integral(ρc * dT/dt * v), is included. This form reduces the derivative order and incorporates boundary flux contributions.
How is the semi-discrete FE form of 2D heat flow derived?
The semi-discrete form uses shape functions to approximate T(x, y) over 2D elements: T(x, y) ≈ sum(N_i * T_i). Substituting this into the weak form yields element equations. For stationary problems, the result is a system of linear equations. For transient problems, a time-stepping method, like the generalized midpoint rule, is applied to handle the temporal term.
What is the discrete FE form of transient 2D heat flow?
The discrete form is a fully algebraic representation of the transient problem. It takes the form [M]{dT/dt} + [K]{T} = {F}, where [M] is the capacity matrix, [K] is the stiffness matrix, {T} is the nodal temperature vector, and {F} represents external loads, including heat flux contributions.
What is the strong form of 2D and 3D elasticity problems?
The strong form of elasticity problems is derived from equilibrium equations, compatibility conditions, and constitutive laws. For 2D, the governing equations are -∇·σ + f = 0, where σ is the stress tensor and f is the body force vector. Similar principles apply for 3D elasticity, with additional components.
What is the weak form of 2D and 3D elasticity problems?
The weak form is obtained by multiplying the equilibrium equations by a test function (displacement variation) and integrating by parts. It reduces the derivative order of stresses. The weak form incorporates essential boundary conditions (displacements) directly and natural boundary conditions (tractions) in the boundary integrals.
How is the FE form of 2D and 3D elasticity problems derived?
The FE form uses shape functions to approximate displacements. Substituting these approximations into the weak form results in element stiffness matrices, load vectors, and the global system of equations. The form also incorporates essential and natural boundary conditions.
What is involved in problem discretization for finite element analysis?
Problem discretization involves dividing the domain into smaller elements, defining nodes, numbering elements, and creating a topology matrix (Edof). A simple mesh is created to represent the domain geometry.
What are element shape function matrices?
Shape function matrices (N) interpolate field variables within elements. For 1D, they are linear or quadratic polynomials. For 2D and 3D, they depend on the element type (e.g., triangular, quadrilateral, or tetrahedral).
What are element shape function derivative matrices?
Shape function derivative matrices (B) contain derivatives of shape functions with respect to spatial coordinates. These are used to calculate strains, gradients, and stiffness matrices in finite element formulations.
What are the element conductivity and stiffness matrices?
These matrices represent the material properties and structural behavior of elements. The conductivity matrix (K) is used for heat transfer problems, while the stiffness matrix is used for elasticity. They are computed through integration over the element using shape function derivatives and material properties.
What should be implemented in FE programs for 1D heat flow problems?
FE programs should solve 1D heat flow problems for all types of boundary conditions (Dirichlet, Neumann, Robin). They should also extend to similar problems like 1D elastic bars and moisture transport.
How should FE programs handle 2D heat flow problems?
FE programs should solve stationary and transient 2D heat flow problems, including essential and natural boundary conditions. The program should accommodate material and geometric variations.
What should FE programs address for 2D elasticity problems?
Programs should solve 2D elasticity problems, incorporating natural (tractions) and essential (displacement) boundary conditions. They should compute stresses, displacements, and reaction forces.
What software can be used to solve 2D FE problems?
Software like Comsol Multiphysics can be used to solve 2D stationary heat flow and elasticity problems. It simplifies problem setup, meshing, and solution processes.
What does convergence of the FE solution involve?
Convergence involves verifying that the solution approaches the exact solution as the mesh is refined (h-convergence) or as the polynomial degree of shape functions increases (p-convergence). Shape functions must satisfy completeness and compatibility conditions.
How can FE problems with non-zero essential BCs be solved?
Non-zero essential boundary conditions are handled by modifying the global stiffness matrix and load vector. The displacement values are imposed directly in the solution process, often through matrix partitioning techniques.
What is the generalized mid-point rule for time stepping?
The generalized mid-point rule is a time-stepping method for transient problems. It is expressed as T_(n+1) = T_n + Δt * θ * f(T), where θ controls the weighting (e.g., θ=1/2 for Crank-Nicholson). Stability and time step size depend on θ and the problem’s properties.
What are suitable boundary conditions from an engineering perspective?
Suitable boundary conditions are those that accurately represent physical constraints, such as fixed temperatures, applied loads, or heat fluxes. The choice depends on the problem and must ensure the solution’s relevance and accuracy.
How can the reasonableness of a given FE solution be assessed?
The reasonableness of an FE solution is assessed by checking physical plausibility, boundary condition satisfaction, and comparison with analytical or experimental results. Convergence studies can further validate the solution.
