All Questions
94 questions
0
votes
0
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128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
2
votes
1
answer
296
views
Asking advice for implementation of Conservative Finite Difference Scheme for numerically solving Gross-Pitaevskii equation
I am trying to numerically solve the Gross-Pitaevskii equation for an impurity coupled with a one-dimensional weakly-interacting bosonic bath, given by (in dimensionless units):
\begin{align}
i \frac{\...
- 21
1
vote
0
answers
50
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
0
votes
0
answers
38
views
Thermo Hydraulic Mechanical modeling of energy wall slab in Comsol multiphysics
I am currently working on a complex simulation project involving an energy wall slab, and I need assistance in accurately modeling and validating it using COMSOL Multiphysics. Here are the details of ...
1
vote
0
answers
70
views
Deriving order of accuracy and interpreting a given discretization scheme when underlying method ( finite difference/volume) not known
If a spatial grid is given with time levels like this:
to solve the following model problem
Now consider the following discretization schemes:
Scheme 1
Scheme 2
Usually, to determine order of ...
- 445
1
vote
2
answers
121
views
How to handle non bilinear weak form?
I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment.
I modeled the ...
- 63
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
0
votes
1
answer
211
views
Using Crank-Nicolson to solve Non-Linear Schrödinger equation in Python
I aim to solve the (non-linear) Schrodinger equation using the Crank-Nicolson method in Python. Here are my two functions.
...
1
vote
1
answer
338
views
Why does scipy Conjugate Gradient solver fail to converge for non-steady heat equation using Crank-Nicolson method
Could someone please explain why my implementation of the Crank-Nicolson method applied to the non-steady heat equation won't converge? There shouldn't be any nonlinear aspects to my implementation ...
- 13
2
votes
0
answers
114
views
Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 2,892
1
vote
1
answer
525
views
Crank Nicolson Method with closed boundary conditions
I want to simulate 1D diffusion with a constant diffusion coefficient using the Crank-Nicolson method.
$$\frac{\partial u (x,t)}{\partial t} = D \frac{\partial^2 u(x,t)}{\partial x^2}.$$
I take an ...
- 111
5
votes
1
answer
107
views
Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
1
vote
1
answer
308
views
2D Heat equation solved with finite element method converges in skewed way
I tried to solve the 2D heat equation with the finite element method, using triangles as elements. Currently generated by a Delaunay triangulation. The base function I'm currently using is basically ...
- 63
0
votes
1
answer
143
views
How to get a normalized gradient with FreeFem++?
I am trying to use FreeFem++ to solve the heat geodesics algorithm.
The algorithm is:
solve $\dot u = \Delta u$ at a specific time $t$.
compute $X = \frac{\nabla u_t}{|\nabla u_t|}$
solve $\Delta\phi ...
- 379
1
vote
0
answers
115
views
Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
0
votes
2
answers
503
views
Can the Crank-Nicolson Method Be used to Solve The Schrodinger Equation with a Time Varying Potential?
I have been following an excellent article about how to use the Crank-Nicolson method to solve the Schrodinger equation. In the article, it starts with a $V(x, y, t)$ but the potential seems to become ...
- 33
2
votes
0
answers
113
views
Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
- 41
1
vote
0
answers
225
views
Solving PDE on a non-uniform grid with Crank-Nicolson scheme
I am solving a 1D diffusion-type equation with the finite-difference Crank-Nicolson (CN) scheme, and I need to densify the spatial grid around the central point. One could change the spatial variable ...
- 21
0
votes
1
answer
148
views
Problem with my Octave code (unsteady heat equation with FEM)
I want help with my Octave code regarding the unsteady heat equation.
My geometry and mesh are generated with FreeFEM++, so there is no problem with that (I tried it with the steady problem with no ...
- 3
1
vote
0
answers
140
views
Crank-Nicolson vs Spectral Methods for the TDSE
The time-dependent Schroedinger equation (TDSE) depends linearly on the system's initial state $\vert \psi(0) \rangle$, such that the solution can be generally written as
$$ \vert \psi(t) \rangle = \...
- 111
2
votes
0
answers
173
views
Error in implementation of Crank-Nicolson method applied to 1D TDSE?
Some context, I've posted this question on physics SE and stack overflow. The former had nothing to offer, the latter had a great commenter that agreed with the phase looking off being one of the ...
2
votes
0
answers
92
views
How to accelerate the computing of implicit finite difference method for heat conduction between two solids
Edit on May 3rd: I have found the problem. Because the difference of between $k_1$ and $k_2$ is huge, a very small time step need to be chosen so that the right green part can "feel" the ...
- 21
0
votes
0
answers
98
views
Transparent Boundary Conditions for Finite Difference ADI PR 2D TDSE solution
I want to put (non-dirichlet) boundary conditions inside the code I wrote to solve the 2dim TDSE using the alternating direction implicit Peaceman - Rachford method.
$$
(1 + iB\Delta t/2 ) \psi^{n+1/2}...
- 141
4
votes
1
answer
220
views
Method to linearize highly nonlinear partial differential equation
I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $T_f$, $T_s$ and $c$ are convection-diffusion-reaction-like ...
- 81
2
votes
0
answers
294
views
Rosenthal equation for multi track
Rosenthal's equation lets one calculate the temperature profile of a moving point heat source analytically for thin and thick plates. For simplicity I use the equation for thick plates defined as:
$$T-...
- 840
2
votes
1
answer
507
views
Solving Schrodinger Equation with finite element and Crank-Nicolson?
I have asked this in Mathematic section, but received no reply.
Please let me ask here to see if threr is any difference.
The Schrodinger equation without potential has the following form:
$$\...
- 119
2
votes
0
answers
163
views
Advection diffusion equation using Crank-Nicolson with total flux and Diriclet BCs
I am trying to model the 1D advection-diffusion equation:
$${\partial c \over \partial t} = D_c{\partial^2 c \over \partial x^2} -u{\partial c \over \partial x}.$$
With Robin boundary conditions that ...
- 81
1
vote
1
answer
2k
views
How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
3
votes
1
answer
170
views
Derivation of a parabolic PDE using Alternating Direction Implicit method
I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme.
If I have an equation of the form:
\begin{equation*}
\frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t)
\end{...
- 319
4
votes
1
answer
139
views
Solving geodesics on triangular meshes gives negative distances
I have implemented the heat method for geodesics:
https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf
When I run it I am getting a solution that, visually, seems correct:
In this image, ...
- 379
5
votes
0
answers
450
views
Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered
Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
- 445
0
votes
1
answer
214
views
Incorporating heat flux into Laplace Equation
I need to find the temperature distribution of a square plate using the Laplace equation by using FDM:
$$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$
But there is a heat flux entering from the top ...
- 145
2
votes
0
answers
89
views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
- 21
3
votes
1
answer
151
views
Maintain unitary time evolution for a nonlinear ODE
I want to solve a nonlinear ODE of matrix $A(t)$
$$\mathrm{i}\dot A = A(t)M(t),\:\mathrm{with}\: M(t)=A^\dagger(t)H(t)A(t)$$ where $H(t)$ and hence $M(t)$ are Hermitian. Therefore, I presume the time ...
- 303
2
votes
2
answers
468
views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
- 61
2
votes
2
answers
219
views
Heat equation in non-dimensional form behaving differently than in usual format
Starting from
$$
c_p \frac{\partial u }{\partial t} = k \nabla^2 u
$$
in a one dimensional domain [0,1] where $c_p$ and $k$ are modeling two different materials:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 601
0
votes
0
answers
103
views
Crank-Nicolson solution of parabolic PDE with Newumann boundary conditions
I am trying to solve the non-linear parabolic PDE in $c(t,r)$
$$c_t=\frac{1}{r}(rDc_r-\alpha r^2 c)_r$$
with initial condition $c(0,r)=f(r)$
and boundary conditions $c_r(t,r_1)=\alpha r_1c_1/D$ and $...
- 171
1
vote
2
answers
811
views
How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly
My post is structured in four parts:
I give you some information about the context my principal questions refer to.
I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
- 11
2
votes
3
answers
553
views
Flux sign and face normal confusion in finite volume method
I implemented a solver for the 2D steady-state heat equation (without heat generation and homogeneous material) $\nabla. (k\nabla T) = 0$, using finite volume method, however, I am having some ...
- 304
0
votes
1
answer
190
views
FDM on nonlinear PDEs
I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$.
In order to perform time discretization with FDM (finite ...
1
vote
0
answers
456
views
Incorporating radiation boundary condition at the edge in finite difference
I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge.
$-k\frac{\partial ...
- 19
3
votes
0
answers
108
views
Correct approach for thermal finite element simulation of layered assembly
I would like to optimise the heat transfer on a PCB. Several dies are on the top and cooling air is going through the fins in heat sink on the bottom. The assembly consists of several layers like ...
- 570
2
votes
1
answer
434
views
Solution method of nonlinear heat transfer analysis
The governing equation of transient heat transfer analysis is described as follows:
$$C \frac{dT}{dt}+K T = Q$$
When using backward difference scheme for the discretization of the time we get the ...
- 840
0
votes
1
answer
51
views
Produce vertex displacements from volumetric shrinkage data on unstructured meshes
I was wondering what would be an efficient way to produce compatible displacements for mesh nodes/vertices if the computed data is volume shrinkage of each element/cell in the unstructured mesh?
...
- 1,413
1
vote
1
answer
349
views
Crank-Nicholson for diffusion-advection vs diffusion equation
Let's consider the following 1D diffusion equation:
$\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$
where we assume that the diffusion ...
- 172
2
votes
1
answer
311
views
Lumped matrices in thermal analysis using finite elements
The governing equation of the transient heat transfer problem is
$$C \frac{dT}{dt}+K T = Q$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
- 840
6
votes
2
answers
408
views
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
I am considering the following diffusion equation:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$
over a grid ...
- 171
1
vote
2
answers
191
views
Simulating the heat equation with insulating material
My plan is to solve the heat equation in the right half portion of the domain, while having the left half completely isolated with constant temperature. To do so, I model the left half with a very low ...
- 601