All Questions
67 questions
11
votes
3
answers
1k
views
Methods of solving non-linear advection-diffusion systems beyond Newton-Raphson?
I'm working on a project where I have two adv-diff coupled domains through their respective source terms (one domain adds mass, the other subtracts mass). For brevity, I'm modeling them in steady ...
- 694
8
votes
1
answer
550
views
Computing geodesic distances with diffusion
I am trying to solve an APSP (All-Pair Shortest Path) problem on a weighted graph.
This graph is actually a 1, 2 or 3 dimensional grid, and the weights on each edge represent the distance between its ...
- 131
7
votes
3
answers
459
views
Strong coupling of a non-linear multiphysic problem: failure with Newton Raphson method
I am trying to solve a multiphysic problem using finite elements and a Newton Raphson solution scheme. I have two non-linear subsystems that are coupled bi-directionally.
The first subsystem includes ...
- 71
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
5
votes
0
answers
103
views
Evolutionary dynamics in vascularised tumors, PDE-ODE coupled system
I have to solve the following PDE-ODE system
$$ \displaystyle{\partial_{t} n = \bigl[a(s) - b(s)(y - h(s))^{2} - d\int_{\mathbb{R}} n \, dy \; + \; \beta \, \partial^2_{yy} n \;}
\\\\
\...
- 61
5
votes
0
answers
198
views
Time-stepping for coupled nonlinear PDEs
What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow),
$$
\begin{cases}
\rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla \mathbf{u}\...
- 3,146
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
4
votes
1
answer
2k
views
How could we solve coupled PDE with finite difference method and Newton-Raphson method?
I'm trying to solve coupled PDE by Crank-Nicolson (CN) and Newton-Raphson method with MATLAB. I have used CN method but not for coupled problem. Please if someone could help let me know to add more ...
- 41
3
votes
2
answers
422
views
Modelling question: example of a physical phenomenon with this jump condition at an interface?
in our finite element class we were talking about interface problems our teacher came up with the following, where $K_i$ are two given functions and $u_i$ is the restriction of the solution $u$ to $\...
- 435
3
votes
2
answers
453
views
Solving coupled PDEs numerically on a semi-infinite domain with no-flux boundary conditions
I have the following system of PDEs for which I have given parameters $\gamma, \tau$ and $\mu$,
$$\begin{align}
T_t = &\ \gamma\,(L +\tau F-T)\\
F_t = & -F_x-(F-LT)\\
L_t = &\ \mu L_{xx}+...
- 31
3
votes
2
answers
384
views
ode45 with matrix initial conditions
EDIT: We have a coupled system of 10 ode each. The coupling presents in the last equation. I thought about using a matrix 10 by 2 as initial conditions.
I also followed a similar question with the ...
- 81
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
3
votes
0
answers
108
views
Correct approach for thermal finite element simulation of layered assembly
I would like to optimise the heat transfer on a PCB. Several dies are on the top and cooling air is going through the fins in heat sink on the bottom. The assembly consists of several layers like ...
- 570
2
votes
3
answers
553
views
Flux sign and face normal confusion in finite volume method
I implemented a solver for the 2D steady-state heat equation (without heat generation and homogeneous material) $\nabla. (k\nabla T) = 0$, using finite volume method, however, I am having some ...
- 304
2
votes
2
answers
468
views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
- 61
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ 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(...
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
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
2
votes
1
answer
134
views
How to couple the vibro-acoustic equations by Mortar method for non-matching meshes?
Assume we have two domains $\Omega_a$ a acoustic domain with boundary $\Gamma_a$ and $\Omega_s$ a domain of a solid body with boundary $\Gamma_s$.
$\Omega_a$ and $\Omega_s$ have the common interface $\...
- 459
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:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 601
2
votes
1
answer
710
views
Solve 3-D Heat equation with Neumann boundaries
I want to solve the Poisson PDE for heat flow in a 3-D solid cube with given dimensions $x$, $y$, and $z$:
$$\rho C\frac{\partial T}{\partial t} = k \Delta T$$
The cube is irradiated with a constant ...
- 21
2
votes
1
answer
244
views
Numerical methods for coupled stiff PDEs
I'm dealing with a set of nonlinear coupled PDEs that have the form:
\begin{align}
\frac {\partial y_1}{\partial t} &= y_2y_3 - y_1 \tag{1}\\
\frac {\partial y_2}{\partial t} &= y_1y_3 - y_2 \...
- 568
2
votes
1
answer
297
views
Coupling Boundary Condition of one PDE with source term of another PDE
We have a system of equations, wherein the BC of one PDE is coupled with the source term of another PDE.
We have a regular 2D unit grid in x and y.
There are two PDEs to be solved
The first PDE (...
2
votes
0
answers
114
views
Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 2,892
2
votes
0
answers
113
views
Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
- 41
2
votes
0
answers
92
views
How to accelerate the computing of implicit finite difference method for heat conduction between two solids
Edit on May 3rd: I have found the problem. Because the difference of between $k_1$ and $k_2$ is huge, a very small time step need to be chosen so that the right green part can "feel" the ...
- 21
2
votes
0
answers
294
views
Rosenthal equation for multi track
Rosenthal's equation lets one calculate the temperature profile of a moving point heat source analytically for thin and thick plates. For simplicity I use the equation for thick plates defined as:
$$T-...
- 840
2
votes
0
answers
83
views
Dynamic Successive Over/under Relaxation (SOR) with several variables
I am solving a partial differential algebraic equation (PDAE) system which has the following dependent variables:
$f=f(X,T)$ and $g=g(T)$, along with a few others
My current method for coupling is ...
- 41
2
votes
0
answers
222
views
Segregated solving of a tightly coupled system of PDEs
To compute the evolution of a free surface between two incompressible, immiscible liquids, two tightly coupled equations have to be solved, the volume fraction advection and the Navier-Stokes ...
- 729
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
1
vote
1
answer
313
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
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
1
vote
1
answer
645
views
Physical interpretation of L2 norm of heat equation solution
For the heat equation
\begin{equation}
u_t(t,x) = \nu u_{xx}(t,x)
\end{equation}
for $x \in [0,1]$ with boundary conditions $u(t,0) = u(t,1) = 0$ and initial value $u(0,x) = u_0(x)$ it is easy to ...
- 1,273
1
vote
1
answer
257
views
Coupled PDE: a confusion in boundary condition setup
I have a coupled PDE problem(Poisson-Schrondinger system), i.e.
first I need to solve an eigenvalue problem (Schrodinger problem discretized by Galerkin method)
$$Ax=\lambda x, ~~~A=A(u)$$
the ...
- 593
1
vote
1
answer
339
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
1
vote
1
answer
15k
views
Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression
Hello all,
I've recently been introduced to Python and Numpy, and am still a beginner in applying it for numerical methods. I've been performing simple 1D diffusion computations. I suppose my ...
1
vote
2
answers
2k
views
Finite Difference Grid Spacing and Scaling
I have been exploring finite differences and heat transfer using the 2D heat equation to further expand my knowledge. So far I think it is going well.
I am running into some confusion around grid ...
- 11
1
vote
3
answers
2k
views
Unexpected results of MATLAB's ode45
Whilst working with MATLAB recently I encountered something odd that I cannot explain. I was using the ode45 solver to solve a system of two coupled second order ODEs. I wasn't convinced about the ...
- 59
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
1
vote
1
answer
56
views
Verification of coupled system of equations for light propagation
I am trying to simulate the propagation of light in material using the non-linear schrödinger equation (NLSE):
$$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp E+\frac{ik_0n_2}{n_0}\vert E\vert^2E-0.5\beta^{...
- 563
1
vote
2
answers
812
views
How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly
My post is structured in four parts:
I give you some information about the context my principal questions refer to.
I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
- 11
1
vote
1
answer
378
views
Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)
I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D.
In order to both test the timestepping and the spatial discretisations I had a look at using ...
- 11
1
vote
2
answers
175
views
Improving calculation algorithm for coupled PDEs
I have the following two PDEs:
$$\partial_zU=\nabla_r^2U+\varrho U$$
$$\partial_t\varrho=a\vert U\vert^4$$
with $a$ a constant and
$$dt=dz\cdot\frac{n}{c}$$ with $n$ the refractive index of a ...
- 563
1
vote
2
answers
399
views
V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference
I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's ...
- 11
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
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
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
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
1
vote
0
answers
68
views
Combining fluid flow solver based on lattice Boltzmann method with a mechanical deformation solver based on finite element method
I'm thinking to couple my fluid flow solver based on lattice Boltzmann method with a mechanical deformation solver based on finite element method to take account for solid deformation in my models. In ...
- 173
1
vote
0
answers
953
views
Methods and tools to solve the two-temperature model (TTM)
I would like to model heat diffusion at the gold / water interface after excitation of the metal surface by an ultrafast laser pulse (ca. 80 fs).
An appropriate model to start with would be the "two ...
- 21