Questions tagged [heat-transfer]

For questions about modeling heat transfer, often through the use of heat (differential) equation.

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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 ...
Boiler4562's user avatar
2 votes
1 answer
281 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(...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
280 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 ...
n1ck94's user avatar
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2 votes
0 answers
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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,\...
lightxbulb's user avatar
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5 votes
1 answer
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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 ...
Dieter Menne's user avatar
1 vote
1 answer
181 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 ...
Boiler4562's user avatar
0 votes
1 answer
92 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 ...
Makogan's user avatar
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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 ...
mathbruh67's user avatar
2 votes
0 answers
106 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 ...
Avrana's user avatar
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1 answer
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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 ...
mibo27's user avatar
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2 votes
0 answers
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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 ...
Kai Jiao's user avatar
2 votes
0 answers
203 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-...
vydesaster's user avatar
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. ...
Alex I's user avatar
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4 votes
1 answer
132 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, ...
Makogan's user avatar
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0 votes
1 answer
201 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 ...
justauser's user avatar
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2 votes
2 answers
391 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 ...
Max_89's user avatar
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2 votes
2 answers
214 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 &...
balborian's user avatar
  • 601
1 vote
2 answers
712 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 ...
tom terenius's user avatar
2 votes
3 answers
470 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 ...
Algo's user avatar
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1 vote
0 answers
388 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 ...
Nitin's user avatar
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3 votes
0 answers
103 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 ...
Ken Grimes's user avatar
2 votes
1 answer
359 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 ...
vydesaster's user avatar
0 votes
1 answer
46 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? ...
Johntra Volta's user avatar
2 votes
1 answer
270 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. $...
vydesaster's user avatar
1 vote
2 answers
180 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 ...
balborian's user avatar
  • 601
0 votes
2 answers
1k views

(FEM) 1D time-dependent heat equation convergence problem

I'm simulating a simple 3-node bar with convection BCs at the edges to validate my FEM code. The following data was used: Initial temperature = 25 ºC Temperature surrounding the rod = 10 ºC Thermal ...
user avatar
0 votes
1 answer
559 views

Heat diffusion - Is this the correct approach to include Newmann boundary conditions?

Thank you for looking at this problem. Is this the correct approach to include neumann boundary conditions? With this solution temperature is not correct, and there´s no diffusion. The model seems ...
M. A.'s user avatar
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1 vote
1 answer
14k 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 ...
Leonardo Parra García's user avatar
0 votes
1 answer
65 views

Modeling Diodes in Autodesk CFD

I'm extremely new to Autodesk CFD, and I'm working on a project that deals with diodes heating up and cooling down based on a fixed temperature regulated by a temperature switch. Basically, I have a ...
Aarthi Muthukumar's user avatar
2 votes
1 answer
606 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 ...
Phillip's user avatar
  • 21
1 vote
1 answer
369 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 ...
mpichael's user avatar
8 votes
1 answer
521 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 ...
matthieu's user avatar
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0 votes
1 answer
79 views

Solving the diffusion/heat equation for a randomly distributed set of points in 3D

In this problem I am trying to solve, I have a messy set of points distributed in 3D space, each with a defined temperature. If I would want to calculate the heat transfer scenario in this system, how ...
Vinícius Godim's user avatar
1 vote
0 answers
877 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 ...
François's user avatar
1 vote
0 answers
203 views

BTCS-like method for heat conduction in unstructured triangular grid

I want to write a simple simulation for heat conduction in a unstructured triangular mesh. I already made it work for a structured rectangular grid with the ADI method, but now I need more complex ...
Hendrik410's user avatar
1 vote
2 answers
387 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 ...
Jeremy Lim's user avatar
1 vote
2 answers
1k 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 ...
Bluebill's user avatar
0 votes
1 answer
807 views

Heat equation with Neumann and Dirichlet conditions on same boundary

I am looking at numerical solutions to the heat equation with Dirichlet and Neumann conditions on the same boundary. That is $u(x,t)$ satisfying $$ u_t = u_{xx}\,, \quad x \in[0,1]\,, \quad t>0\,,...
Steve's user avatar
  • 541
1 vote
1 answer
610 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 ...
Daniel's user avatar
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