Pihua Wen Flashcards
Why do we need to consider the Cauchy Principle Value?
In the standard equation, we are given too many unknowns. We then move the source point to the boundary to get another input, but this causes a singularity when r = 0, due to the fact that the fundamental solutions depend on 1/r. CPV are used for improper integrals like this, and we use C(x’) = 0.5 for a smooth boundary.
List two advantages for the Finite Element Method and the Boundary Element Method
FEM:
Ad:
- Very easy to generalise to arbitrary cases
- Convergent
- Many commercial packages available
Dis:
- Coarse mesh results in poor results (no problem with high density of elements)
- Can’t deal very well with singularities such as cracks. Point of tiny areas have infinite stress.
BEM:
Ad:
- Very high accuracy
- Dimension reduction. Means 3D can be treated as 2D etc. significantly reducing computational requirements.
Dis:
- Need fundamental solution to be available
- Not easy to generalize, for example, non-linear or dynamic problems require special treatment.
How may boundary values at each boundary node for potential problems?
2 - Heat and Flux
How could we determine boundary unknowns?
Apply linear interpolating shape functions to the BIE, then move knowns to LHS and unknowns to RHS in the form Ax = z and solve for unknowns.
What is the Boundary Element Method?
numerical computational method of solving linear partial differential equations which have been formulated as integral equations, which uses interpolating elements applied to the Boundary Integral Equation
Describe the steps in the derivation of BIE
Div thm -> Greens second identity -> BIE
What is a boundary condition?
A condition that is required to be satisfied at all or part of the boundary of a region in which a set of differential conditions is to be solved
What are fundamental solutions, and why do we use them?
A fundamental solution(u* or q) is a solution to the dirac delta function as a function of the source point and a field point. They are a key part of the BIE, and have known equations. We use them because the u and q* are arbitrary, but in selecting these specific fundamental solutions, we can introduce two unknowns which only depend on boundary conditions, allowing us to solve for properties within the boundary using only boundary information.
How can we measure temperatures in the domain using BEM?
Apply elements to BIE. We can numerically solve using interpolating elements applied to BIE, then input the internal coordinates into the fundamental solutions, apply the boundary conditions and solve.
Why are boundary elements required? What type of element should be used for a 2D problem?
Elements are used to discretize one arbitrary problem, into many smaller general problems. A 2D problem can use 1D boundary elements.
Why is BEM useful in contact mechanics?
Because contact mechanics problems are usually non-linear, which makes obtaining data about a given scenario very difficult in the absence of experimental data, which is often extremely difficult to obtain. Consequently, it typically requires numerical methods.
Describe the 3 modes of contact and the corresponding Coulomb equation.
Separation - no contact, not friction,
Slip - sliding - F_f = mu*Fn
Stick - no movement - unknown F_f - very difficult.
What are conforming contacts?
Those with the same surface configurations i.e. completely flush.
What are progressive and recessive contacts?
Progressive contacts increase contact area with time, recessive contact areas recede with time.
Why do we need multiple contact zones for some problems?
Some problems, such as non-conforming ones, need 3 contacts due to the need for 4 equations to determine variables. Non-linear problems do, however, get introduced to the problem of not knowing where the contact zones end. This typically requries iterative analysis, as they are dependent on the objects history.
