All Questions
112 questions
2
votes
1
answer
87
views
Integration involving bessel functions
I want to (efficiently) integrate a function of the form
$$
\int_0^a f(r) J_1(kr) \, dr,
$$
where $f$ is some function which involves trigonometric functions and where I do not always have $a,k\gg 1$.
...
- 21
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.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
CommunityBot
- 1
0
votes
0
answers
128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 1
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 ...
- 447
0
votes
0
answers
63
views
Are there any efficient numerical methods to solve the recurrence relation of a function？
First, the recurrence relation is:
$$\pi_{k+1}(\omega)=(\omega-\alpha_k)\pi_{k}(\omega)-\beta_k\pi_{k-1}(\omega),$$
where
$$\alpha_k=\frac{\int_0^\Lambda\omega\pi_{k}^2(\omega)h^2(\omega)\text{d}\...
- 31
0
votes
0
answers
36
views
Fastest way to calculate the eigenvector with the largest eigenvalue for a 3*3 positive-definite matrix [duplicate]
As stated in the title: I have a 3 by 3 positive-definite matrix $M$. What I need is the eigenvector corresponding to the largest eigenvalue, since I am calculating the solution to maximize the value ...
- 101
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
1
vote
0
answers
50
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
1
vote
2
answers
121
views
How to handle non bilinear weak form?
I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment.
I modeled the ...
- 63
2
votes
1
answer
453
views
How to leverage the GPU for parallel 3-body problem computations
I have a 3-body simulation which must run millions of times.
As far as I know, the GPU shines when it gets to preform simple operations on huge matrices/arrays. Currently I'm debugging and running my ...
- 41
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ 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(...
- 1,498
5
votes
0
answers
152
views
Single precision vs double precision conjugate gradients
I tested my conjugate gradients implementation with float and double precision and contrary to my guess the double code was twice faster than the single precision code. The reason is that I need many ...
- 2,892
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 ...
- 2,892
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
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 ...
CommunityBot
- 1
8
votes
1
answer
751
views
Accurate and efficient computation of the inverse Langevin function
The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ occurs in computations related to elastomers and paramagnetic materials. It is easily computed accurately and with high ...
- 1,895
1
vote
1
answer
309
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 ...
- 8,622
2
votes
1
answer
93
views
Best approach to simulating dynamics on networks
I have been recently getting into the field of various processes on networks. For example, stochastic processes like percolation, Ising models, various statistical-physics models; or deterministic ...
- 2,147
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 ...
- 3,101
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
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
3
answers
197
views
Inefficient comparisons of custom data type C++
I've got some code that I need to squeeze every bit of both time and space out of. I'm looking for a better solution to the following problem.
For reasons outside of the scope of this question, I ...
- 2,345
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 ...
- 1,973
5
votes
0
answers
109
views
Dense factorization specialized for RBF-FD method
In RBF-FD methods (see Fornberg & Flyer. A Primer on Radial Basis Functions with Application to the Geosciences. SIAM, 2015. Chapter 5.), the finite-difference stencil coefficients for a set of ...
- 645
1
vote
1
answer
146
views
Computing pairwise distances in a grid efficiently
I have a spline-based model where I have a set of control points in 3D space. I want to compute pairwise distances between every point on a regular 3D grid and these control points. Is there an ...
- 654
1
vote
1
answer
168
views
Efficiency of developing PDE solvers using sparse matrices versus loops
I am new to solving PDEs, but have been looking at different implementations of finite difference and finite volume schemes. One thing I have noticed in different implementations is that some ...
- 3,065
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-...
- 1,023
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.
...
CommunityBot
- 1
17
votes
2
answers
1k
views
When is automatic differentiation cheap?
Automatic differentiation allows us to numerically evaluate the derivative of a program on a particular input. There is a theorem that this computation can done at a cost less than five times the cost ...
- 1,023
4
votes
1
answer
727
views
Incomplete Cholesky preconditioner for CG efficiency
I am currently solving the harmonic equation using a P1 FEM discretisation. The resulting matrix $A$ is SPD and fairly sparse so I use a preconditioned conjugate gradients (CG) solver to find a ...
- 2,892
-1
votes
1
answer
98
views
How asymmetric encryption is done very fast [closed]
In any asymmetric encryption specifications, there is a step where we need to calculate data ^ public_key mod e to get ...
- 2,851
2
votes
1
answer
412
views
Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
- 1,124
3
votes
1
answer
541
views
How can I extract the banded or block diagonal part of a sparse matrix in MATLAB?
Given a large sparse (square) matrix in MATLAB, how can I extract the banded or the block-diagonal parts (of fixed size) of it efficiently?
These are useful operations when prototyping and testing ...
- 158
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, ...
- 8,622
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$$
But there is a heat flux entering from the top ...
- 56.8k
2
votes
2
answers
468
views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
- 10.9k
2
votes
2
answers
219
views
Heat equation in non-dimensional form behaving differently than in usual format
Starting from
$$
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 &...
- 1,335
2
votes
1
answer
303
views
Methods to improve the efficiency and the memory requirement of LU factorization for complex symmetric system matrix
I want to solve a linear set of equations (Ax=b) using LU decomposition. My "A" matrix is a complex matrix which is ...
- 5,076
2
votes
1
answer
273
views
Difference between Brent's and Alefeld-Potra-Shi for root finding
I need to find the (unique) root of a nonlinear function $f(x)$, $x \in \mathbb{R}$.
For the record, $f(x)$ is the CDF of a probability density minus a constant $0 < p < 1$ (I am inverting the ...
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
1
vote
1
answer
706
views
Effecient method for iterating over sparse dataset
Apologies if this isn't the appropriate forum for this question.
I have a set of elements that I need to iterate over as part of a modeling workflow. The elements exists over a set of dimensions (i, ...
- 56.8k
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 ...
- 1,498
3
votes
1
answer
141
views
How to justify using available code (in different language) for comparing algorithms
I have proposed an algorithm for a scheduling problem in a submitting paper. In the revision, the reviewer asked us to compare with another algorithm from the literature. Our algorithm is in MATLAB, ...
- 133
0
votes
0
answers
69
views
How can I optimize that loop?
I need to populate a matrix $A_{kl}$, where
$$ k = (m-1)J+n$$
$$ l = (p-1)J+q$$
And
$$m,p = 1, 2, ..., I$$
$$n,q = 1, 2, ..., J$$
Its components are (mnpq). For ...
- 294
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 ...
- 2,859
5
votes
1
answer
870
views
Fastest way to calculate the $2$-norm (or an upper bound for the $2$-norm) of the inverse of a matrix $A\in \mathbb{C}^{N\times N}$
I have a matrix $A\in \mathbb{C}^{N\times N}$ and I need to calculate $||A^{-1}||_{2}$ efficiently. Can it be done without having to evaluate the inverse explicitly?
In general, I am looking for ...
- 8,742
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?
...
- 2,332