10
votes
Accepted
How do I simulate an open end?
The problem you describe, how to prescribe non-reflecting
or absorbing boundary conditions when solving partial differential
equations (PDE) has been extensively studied. For complex (e.g. nonlinear)
...
- 6,229
10
votes
How to solve a second order differential equation (diffusion) with boundary conditions using Python
I have found that I must keep the value of dt near dx or the results become unstable
This behavior you have noticed is known as the Courant–Friedrichs–Lewy (CFL) condition. Indeed, there are ...
- 1,124
8
votes
Accepted
Symmetric boundary condition
For the sake of simplicity let me slightly change your notation. Let $u, v, w$ be the components in the $x, y, z$ directions of a true (polar) vector, and $y=0$ the symmetry plane. For a true vector, ...
- 3,839
8
votes
Accepted
Solving Poisson equation with current BC using FEM
It is standard procedure for general case like that. You can enforce condition for electrode by Lagrange multipliers,
$$
L(u,\lambda) = \int_\Omega \sigma u_{,i} u_{,i} \, \textrm{d}V - \int_{\...
- 906
7
votes
How can an engineering student become a computational scinece expert in a short time
There is no shortcut. Just like there is no shortcut to becoming an "engineering expert in a short time".
The thing is that to be an expert in civil engineering, you need to understand load analysis, ...
- 56.2k
6
votes
Accepted
Boundary condition for Pressure in Navier-Stokes equation
When we say that the pressure is only defined "up to a constant", what we mean is this: If $u,p$ is a solution to the Navier-Stokes equations, then $u,p+c$ for any constant $c$ is also a solution. In ...
- 56.2k
6
votes
Computing accurate fluxes with FEM
If $\nabla u$ is of more interest than $u$ itself, it is reasonable to set $\mathbb v := -\sigma \, \nabla u$ and convert your Poisson problem
$$
-\nabla \cdot (\sigma \, \nabla u) = f \quad \text{in }...
- 901
6
votes
Accepted
How to apply non zero Dirichlet boundary condition in finite elements?
Consider the fact that your linear system can be broken up into two parts: the equations that correspond to the known Dirichlet dofs, and the equations that correspond to the unknown dofs. For the ...
- 1,512
6
votes
Accepted
FEM current toy problem
The formulation of this problem is tricky. Here is what you have in your original post:
Find $h \in L^2(\partial D)$ such that for any $w \in H^{\frac 1 2}(\partial D)$,
$$
\int_{\partial D} hw \,...
- 56.2k
6
votes
Accepted
Traction -> stress; stress->displacement gradient
In short, no. Neumann boundary conditions should be specified in terms of traction. This is clear when you move from a strong-form statement of the boundary value problem to a weak-form.
Strong form:
...
- 1,512
6
votes
Modelling question: example of a physical phenomenon with this jump condition at an interface?
$\vec\Phi = K_i \nabla u_i$ is the flux across the interface. For example, if $u$ is the thermal energy density and $K$ the thermal conductivity, then $\vec\Phi$ is the thermal energy flux. Energy ...
- 56.2k
6
votes
Starting configuration for Molecular Dynamics
Usually one needs to employ periodic boundary conditions (at least in the horizontal directions). Any atoms which fly outside of the box will be mapped to the opposite side. This also has to be ...
- 3,005
5
votes
Poisson equation finite-difference with pure Neumann boundary conditions
I know this question is old but I found an error that I wanted to clear up in @DrHansGruber 's answer. It took me a few hours to spot the issue and I wanted to post the solution to save anyone else ...
- 101
5
votes
Accepted
Impose Neumann Boundary Condition in advection-diffusion equation 1D
First, write out the semi-discrete equation for $u_0$, assuming it's an interior node:
$ \frac{d}{dt}(u_0) = - c \frac{\left( u_1 - u_{-1} \right)}{2h} + \alpha \frac{\left(u_1 - 2u_0 + u_{-1}\right)}...
- 343
5
votes
Accepted
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
Yes, the problem with mixed boundary conditions is well posed. What's not clear to me is this:
Why do you approximate the derivative via the two-sides approximation? Shouldn't it be enough to just ...
- 56.2k
5
votes
Accepted
Numerical solution of zero-potential time-dependent Schrödinger equation in 1D
Regarding the boundary conditions: Don't be fooled by Wikipedia. Yes, the scenario in the picture suggests an absorption at the boundaries, and yes, one could use absorbing boundary conditions in ...
- 3,197
5
votes
Accepted
Imposing no flux boundary conditions on variants of the Cahn-Hilliard equation using Finite Differences in Python
The short answer is that you need
$$\phi_{-1} = \phi_0$$
$$\phi_N = \phi_{N-1}$$
to impose $\nabla\phi=0$.
A quick check by making the following change
...
4
votes
Solving a linear equation system with pure Neumann condition
Including even one Dirichelt condition changes the problem you are trying to solve and will not give you the correct solution! You must fulfil the Discrete Compatibility Criteria, see e.g. first and ...
- 127
4
votes
Solving the Poisson equation with Neumann Boundary Conditions - Finite Difference, BiCGSTAB
One of the standard ways to handle singular problems like Poisson with Neumann boundary conditions which lead to a sinuglar problem $Ax=b$ is to make the obtained solution orthogonal to the kernel (...
- 254
4
votes
How to incorporate the boundary conditions with the Galerkin method?
Here is a method known as basis recombination, which has not been mentioned in the present thread. I'm citing from the book of J.P. Boyd, "Chebyshev and Fourier Spectral Methods", 2nd Ed., Chapter 6.5....
- 3,197
4
votes
Accepted
$O(h^2)$ convergence for Elliptic PDE
You appear (and correct me if I am wrong, i do not wish to presume) to be confused on the notion of what is an order of convergence.
The order of convergence is the order at which your approximate ...
- 1,157
4
votes
Accepted
Initial Condition in a Numerical Problem
That depends on the equation you have, and on the situation you want to model. Imagine, for example, that you are considering the advection equation
$$
\partial_t u + c \partial_x u = 0,
$$
i.e., ...
- 56.2k
4
votes
Accepted
Computing accurate fluxes with FEM
If averaging the element fluxes, as suggested by @Bill Greene, does not produce accurate enough fluxes, there are ways to improve the accuracy of the fluxes by post-processing. A standard and quite ...
- 482
4
votes
Solving Poisson equation with current BC using FEM
Like @likask points out in the other answer, using Lagrange multipliers is one way to do this. The other is to recognize that if you have constraints of the form $\sum_i a_i U_i=0$, then you can ...
- 56.2k
4
votes
Accepted
Line integral along the edge of an isoparametrically mapped triangle
Interestingly enough, you are using quadrature rule for a master triangle in order to integrate over a segment. You should use quadrature rule for a master segment (e.g. $[-1, 1]$) instead. You should ...
- 901
4
votes
How do you apply boundary conditions in a time-stepping problem?
As HBR mentioned, the boundary conditions can often be immediately incorporated into $A$ and $b$. For example, suppose we wish to solve the 1D heat equation with Dirchilet boundary conditions
$$
u_t =...
- 380
4
votes
Accepted
Structural mechanics - traction free = Zero displacement gradient?
The traction is defined as
$$
\mathbf{t} = \mathbf{n} \cdot \boldsymbol{\sigma}
$$
In terms of components, the zero-traction condition is
$$
t_j = \sum_i n_i \sigma_{ij} = 0
$$
From the above ...
4
votes
Accepted
Radiation boundary condition (heat transfer)
This is not particularly difficult once you realize that the nonlinear boundary condition simply yields a nonlinear term in the weak formulation. Let's assume that you want to solve the steady state ...
- 56.2k
4
votes
Why naively chopped finite difference matrix works for different ODE boundary conditions
If you convert the first line of your matrix operator back into a difference equation you get $$2y(x)-y(x+h)=h^2b,$$
where $b$ is whatever is on the right hand side.
This is consistent with a non-...
- 2,269
4
votes
Applying Neumann boundaries to Crank-Nicolson solution in python
Boundary conditions are often an annoyance, and can frequently take up a surprisingly large percentage of a numerical code. How to implement them depends on your choice of numerical method. Finite ...
- 2,269
Only top scored, non community-wiki answers of a minimum length are eligible