New answers tagged computational-physics
1
Whether any mesh updating (i.e. re-meshing) is required as part of the
solution of the nonlinear equations depends very much on the specific
problem you are trying to solve (geometry, loading, material properties).
Many problems of this type can be solved with no re-meshing.
For example, if you take a block of rubber, create a uniform mesh of
rectangular ...
2
If you are familiar with einsum, maybe this explanation does it: axes[0] and axes[1] specify the locations of the repeated letters in the parameters of einsum. For instance,
np.tensordot(a, b, axes=[(0,2),(3,1)])
corresponds to
np.einsum('ijkl,mkni', a, b)
Indeed, 'ijkl'[(0,2)] == 'ik' == 'mkni'[(3,1)], and all the other letters are distinct.
1
There are few issues in your code. The main one is that, in the P(x) function, return np.exp(-(beta*E(x))) should be return np.exp(-(beta*x)).
Then you should increase the maxdr value. I reckon a value of 0.01 should be enough. These two changes are enough to get a plot that looks realistic.
A couple of additional points:
In general, attempting to move ...
0
As far as I can see, you are not applying any bias to your simulation. The only place where the bias Q comes into play is in the calcEnergy method, which is never called by your code.
In general, the bias need to be used in the actual MC code when you compute the acceptance probability of your move.
7
Your lattice consists of 5 x 5 x 5 = 125 spins, so your number of Montecarlo steps to reach equilibrium should be >> 125, because you randomly picking a site and flipping it, so random numbers should uniformly generated so that it will cover whole lattice. For much finer measurement of thermodynamic quantities, you should take more number of points between ...
5
I sincerely thank @Daniel Shapero for directing me towards this answer.
Discontinuity in the specific heat or susceptibility curves to be visible significantly, you should take much more finer measurements for large number of sweeps, say, I ran the simulation for
1024 steps for system to reach equilibrium
1024 steps for sparse averaging/to measure specific ...
1
I think your approach is very good. Several studies show that doing something stays deeper in memory than reading about it and testing things for yourself will give you a good intuitive feeling. Now, to your questions:
Is MATLAB capable of all this? Yes, it has strong numerical and GUI capabilities
If not, what is? As mentioned by Abdullah Ali Sivas, you ...
2
If movement is removed there, deformation of the ring under variable load remains. This can be calculated in the model of an elastic body with the addition of Rayleigh damping. The animation shows the deformation of a rubber ring calculated using FEM and Mathematica 12.
3
It seems that the type of algorithms differ considerably depending on whether the problem is:
Quasistatic elastic or
Hyperelastic
In the quasistatic elastic case, a simple approach is the following:
As the first part of each timestep, the displacement field $u$ is computed.
Since the displacement $u$ is now known at each node of the mesh, the nodes can be ...
1
TL,DR: either the good old Galerkin finite element method, or mesh-free / particle methods.
There are a few things to unpack here.
First, the simulation you show includes contact between an elastic body (the ring) and a hard boundary, so the constraints are non-holonomic.
Contact problems are much more challenging than, say, elastic deformation under ...
4
You have to write your second order equation as a system of two first order equations. Let $y' = v$, then your equation
$$
y'' + \omega^2 y = 0
$$
becomes
$$
\begin{pmatrix} y' \\ v' \end{pmatrix} = \begin{pmatrix} v \\ -\omega^2 y \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -\omega^2 & 0 \end{pmatrix} \begin{pmatrix} y \\ v \end{pmatrix}
$$
If you ...
1
I can describe the simplified version of this algorithm which might help you to understand what's going on here. Let's you have this Time-dependent Schrodinger equation:
$$i \hbar \partial_{t} \Psi = -\frac{\hbar^{2}}{2m} \nabla^{2} \Psi$$
For a moment forget about potential cause that complicate things here, which might be necessarily helpful for ...
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