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 ...
MPIchael's user avatar
  • 2,852
4 votes
Accepted

Dirichlet condition in finite element method

From a different perspective, regardless of where a system $A\cdot x = 0$ comes from (does not have to be FEM), if you change your mind and would like to prescribe part of $x$, then you'd necessarily ...
Mikael Öhman's user avatar
4 votes
Accepted

How to evaluate the points near/at the boundary when using Richardson extrapolation for improved accuracy of a derivative

Your question was quite interesting - I haven't seen it addressed in any of the books on FDM that I have read (LeVeque, Morton, Lapidus, Thomas). I had never thought about this, so I tried to come up ...
lightxbulb's user avatar
  • 1,994
3 votes
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Non-standard boundary condition for incompressible Navier Stokes

I managed to resolve the issue, Need to compute the values only for the inner part of the domain, then apply BC for the boundary values. The only issue is that there is a backward scheme at the ...
2Napasa's user avatar
  • 362
3 votes

Symmetry axis boundary condition

Since your problem is axisymmetric, your resultant flowfield does not depend on the angle. However, it depends on the radial and longitudinal coordinates if you are solving a stationary flow problem. ...
Nikola Ristic's user avatar
3 votes

Starting configuration for Molecular Dynamics

The biggest issue in your subsequent simulation won't be the periodicity of the box -- it will be trying to equilibrate your system down from what looks like a tangle of lipids into an orderly ...
Shern Ren Tee's user avatar
3 votes
Accepted

Solving Poisson's Equation with Periodic Boundary Conditions

Here's what I think the problem is. You have the equation: $$\Delta u(x) = f(x), \, x\in \Omega.$$ where $\Omega$ is a circle/torus. First you have some compatibility constraints you have to satisfy. ...
lightxbulb's user avatar
  • 1,994
3 votes

Crank Nicolson Method with closed boundary conditions

I tried to run your code and I guess there might be a small mistake in your derivation, When you impose $u_{-1}^{j+1} = u_1^{j+1}$ into the equations at the ends, you have to equate $u_{-1}^{j+1} = ...
RandomElasticity's user avatar
3 votes
Accepted

Why is the maximum potential energy greater than the maximum kinetic energy?

You've defined your potential energy incorrectly. Your spring forcing function is $$ F = b x + w^2 x^{p-1} $$ The change in potential energy is defined as $$ \Delta V = \int_{x_0}^{x_1} F dx = \left.\...
helloworld922's user avatar
2 votes

How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

Here's my justification for setting the nodal values to $g(\mathbf{x})$. There might be some details missing because this is an online answer and not a math paper. Let's start with the homogeneous ...
jeremy theler's user avatar
2 votes

Dirichlet condition in finite element method

From the finite element theory, if you specify Dirichlet boundary condition on part of your domain, then the test function is zero on that part. This removes the effect of any force applied both ...
RandomElasticity's user avatar
2 votes

How to impose boundary conditions when solving a nonlinear dynamical system given by the FEM solver

The general approach in a Newton method is to solve for increments $\delta u$. Then, if your current solution already has the right values at the boundary, $\delta u$ needs to have zero boundary ...
Wolfgang Bangerth's user avatar
2 votes
Accepted

What is the correct way to implement Neumann type boundary conditions for solving a PDE using Chebyshev's collocation method?

Spectral methods are global: Modifying a single value automatically changes the whole approximation If you force the Dirichlet BC on the left side you unfortunately change the derivative (Neumann BC)...
ConvexHull's user avatar
  • 1,290
2 votes
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Symmetrization of Laplacian Matrix Operator (finite volumes)

I finally figured it out. We need to move to the new variables. (Idk why the typeset result formulas look different than the preview.) Let us have $m$ intervals in $x$ direction and $n$ intervals in $...
2Napasa's user avatar
  • 362
1 vote

Dirichlet condition in finite element method

After some studies, I want to provide an example of a case where it can be shown that the submatrix $K$ as described in my question is actually positive-definite, not just positive-semidefinite. ...
Jake1234's user avatar
  • 145
1 vote

First derivative central differences with reflecting boundary conditions

The factor 2 arising in the approximation of $\partial_x u_1$ and $\partial_x u_{n+1}$ is correct. It is correct because you are discretizing your boundary conditions ($\partial_x u_0$ and $\partial_x ...
Sthavishtha Bhopalam's user avatar
1 vote

How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?

(I am not a mathematician, so there might be some inaccuracies) To bring your question in a broader context, note the following: Your system can be written as $$L(u,u',u'',x,t)=0$$ on $\Omega$ with in-...
Bort's user avatar
  • 1,285
1 vote
Accepted

How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?

For Dirichlet BCs, this process is probably a bit simpler than you are thinking. I will demonstrate the procedure for a 1D Poisson problem, but it translates very easily to other problems with ...
whpowell96's user avatar
  • 2,259
1 vote

How can we use the two boundary conditions for the Taylor-Maccoll Equations for Cone Shock Waves?

The traditional way of doing this was to start from the shock with a free stream Mach number of M and an arbitrary shock angle, then march inward until you encountered radial flow at some value of $\...
Philip Roe's user avatar
  • 1,154

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