All Questions
318 questions
2
votes
0
answers
71
views
Can I change the coordinates of the problem to avoid dynamic mesh?
I am looking to simulate the velocity field of a fluid around a fish. Fortunately, some analytical functions have been derived for specific cases (e.g., A Generalized Slender-Body Theory for Fish-Like ...
- 21
1
vote
0
answers
97
views
Can boundary condition be implemented as source term?
TL;DR: Can boundary condition be implemented as source term? Are they equivalent? This post is long because of its context and background.
I'm trying to simulate electrohydrodynamic flow over a domain....
- 11
2
votes
1
answer
125
views
Solving advection-diffusion equation on non-rectangular domain
I am trying to solve a PDE similar to the advection-diffusion equation:
$$
\frac{\partial T}{\partial t} + (\vec{u} \cdot \nabla)T = D \Delta T
$$
(where $\vec{u}$ is a known advecting vector field) ...
- 121
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
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
0
votes
0
answers
62
views
What is the best finite volume method for the following equation?
I'm trying to create a partial differential equation that approximates 1-D climate in a rocky planet's atmosphere, which accounts for energy transport via radiation and convection. I am only ...
- 317
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
1
answer
101
views
Moving least square method in finite volume method
Consider a differntial equation like : $\nabla .(\nabla u)=cte$, using finite volume method we can write $\int\nabla .(\nabla u) dv =\int n.(\nabla u) dA $. Here we want to use the moving least square ...
- 33
2
votes
0
answers
71
views
Lowest order Raviart Thomas elements
I have questions regarding the implementation of Lowest order Raviart Thomas elements on quadrilaterals, some papers use the basis functions in this form:
for example the right edge: $N = \left[\dfrac{...
0
votes
0
answers
50
views
Finding neighboring cells using Gmsh API
I am creating a simple mesh on a square domain, $[-5,5]\times[-5,5]$ using the following .geo file
...
- 183
0
votes
1
answer
128
views
Need help with adaptive meshing code
I am trying to understand adaptive meshing and is using this code (https://github.com/esquivas/amr1d) as a reference. However, there is no documentation for it and thus, it is hard for me to ...
- 101
0
votes
1
answer
61
views
numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?
I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
- 445
0
votes
1
answer
121
views
Generating unstructured finite volume mesh
I want to generate a triangular mesh over a rectangle domain in order to solve Euler equations. Most mesh generator generate a mesh while providing node connectivity for each element. This is ...
- 57
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
104
views
Creating nonuniform grids for FDM with multiple points of concentration
If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use:
$$
S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S
$$
where $c=\...
1
vote
0
answers
60
views
Finite volume method for a general flux
How to approximate flux 𝐹(𝑢)⋅𝑛 where 𝑛 denotes the unit normal outward when using finite volumes?
in my case it's not a conservation law so my question is how can we approximate the final term
\...
- 21
1
vote
1
answer
182
views
How to approximate the flux when using finite volumes?
How to approximate flux $F(u)\cdot n$ where $n$ denotes the unit normal outward when using finite volumes?
$$\int_{\sigma} F(u) \cdot \boldsymbol{n}_{K, \sigma} \mathrm{d} \gamma(x)$$
- 21
3
votes
1
answer
179
views
Non-standard boundary condition for incompressible Navier Stokes
I am having difficulties applying the boundary condition
$$\frac{\partial \vec{V}}{\partial t} + u\frac{\partial \vec{V}}{\partial x} = \frac{1}{\operatorname{Re}}\frac{\partial ^2 \vec{V}}{\partial y^...
- 362
4
votes
1
answer
161
views
Burger's equation (PDE) does not work with downwind difference?
I'm working on implementing the discretised Burger's equation. I am quite confused as to why it does not work when using a step-function and downwind difference formula. When using a step-function and ...
- 43
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ 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(...
4
votes
0
answers
129
views
FVM for non-regular domain with triangular mesh
Setup
The 1D convection-diffusion equation is given by:
\begin{equation}\tag{1}
\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} - \mu \frac{\partial^2 u}{\partial x^2} = 0,
\end{...
- 91
1
vote
1
answer
124
views
Adding a diffusion term to the MUSCL - Kurganov and Tadmor central scheme
Im currently using a MUSCL scheme with a rusanov flux and Van Leer limiter to simulate the 2d euler equations:
$$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho v_x}{\partial x} + \frac{\...
- 13
2
votes
3
answers
251
views
Approximation of derivatives in the finite volume method
I am looking into the finite volume method and I have come to a problem with discritisation. Suppose I am looking at a particular cell(I'm dealing) with cartesian grid. at the points $(X_{i},Y_{j}),(...
- 131
2
votes
2
answers
143
views
How is entropy taken into account in a FV solver?
When solving for a given PDE, how is the entropy taken into account? Does the fluxes has another form? Or is the entropy PDE for a given entropy-flux included in the solver?
- 57
2
votes
1
answer
104
views
Symmetrization of Laplacian Matrix Operator (finite volumes)
The aim is to construct symmetric Laplacian matrix $L$ for 2D square domain.
I have the following discretization of second derivatives on non-uniform grid (will skip the steps of derivation, any ...
- 362
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
0
votes
0
answers
32
views
Parallel Block-Structured class abstraction for FDM
I’m currently developing a FDM/FVM (using contravariant coordinates) code using Fortran and Co-Arrays (SIMD, in general), and so far I have all sparse matrix (BiCGStab, working on AMG) solvers and ...
- 251
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
0
answers
42
views
How to find volume of a depression in a bone
This is known as an articular pillar. It lies on the outer surface of the bone. We have 3D Slicer openCV, but we are unable to find it as its a little irregular in shape and also its 3D in nature. ...
- 1
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
3
votes
2
answers
252
views
Non-conservative advective term in a finite volume scheme
I am interested in solving this set of nonlinear couples advection-diffusion equations using a finite volume scheme:
$$
\frac{\partial f(x,y)}{\partial t}=-(\boldsymbol{u}+\nabla\eta)\cdot\nabla f +\...
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
1
answer
210
views
Non-Uniform Grids: Approximation Quality: First Order Finite Difference vs. First Order Finite Volume
Consider the advection/transport equation in 1D with constant velocity $a(x) \equiv 1$
$$u_t(t,x) + u_x(t,x) = 0$$
on a, say, periodic domain.
On uniform grids $$ \{x_i\}_{i = 1, \dots, N}, \quad x_{i ...
- 1,093
2
votes
0
answers
113
views
Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
- 41
1
vote
0
answers
101
views
Fortran - Lid-Driven Cavity Boundary Conditions Error when using SIMPLE method
I am studying Numerical Methods for incompressible flows. part of the tasks is to model the lid driven cavity problem in 2D using the SIMPLE method.
I have been provided with Fortran code that is ...
- 21
1
vote
0
answers
66
views
Exponential Integrator to solve PDE with Stiff term
I wish to solve an equation like the following,
$$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x}\left(A(x)f\right)=S(x,t)f$$
where $A(x,f)f$ and $S(x,t)f$ are the advection and the source ...
- 97
2
votes
1
answer
135
views
Numerical solution of an advection equation, $\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$, with finite volume
I was trying to solve the following equation numerically,
$$\frac{\partial P}{\partial t}+\frac{\partial}{\partial x}\left(P^{5/3}\right)=0$$
I adopted the Godunov approach for discretising the ...
- 97
0
votes
1
answer
206
views
Gmsh Python: Specify mesh regularity conditons
I am using python API of Gmsh to generate a mesh for a rectangular domain. I am really new at this. My code looks like this,
...
- 183
1
vote
0
answers
91
views
(Algorithmic) Differentiation capable Finite Volume Software: Generation Jacobian
I am looking for Finite Volume Software that employs a method-of-lines like approach by constructing from the hyperbolic PDE of form
$$\partial_t \boldsymbol u(t,\boldsymbol x) + \nabla \cdot \...
- 1,093
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
0
votes
0
answers
68
views
Can I use MOL to solve 2D steady state PDE in terms of r and z spatial coordinates?
recently I need to solve a 2D steady state PDE equation.
It’s not time dependent, and the only two independent variables are z and r direction.
So far for this solution, I was thinking using Method ...
- 1
1
vote
1
answer
99
views
How to solve advective equation with source term depending on variable
I have the following equation
$$
\dfrac{\partial s}{\partial t} + \nabla \cdot \left( \vec{v} s\right) = f(s)
$$
Where $f(s)$ is an explicit source term that depends on $s$, e.g., ($\sin(s)\;cos(s)$).
...
- 21
2
votes
0
answers
80
views
Centered finite volume scheme for an advective term on an unstructured/irregular/non-uniform grid
Consider the continuity equation
$$\frac{\partial u}{\partial t} + \frac{\partial \Phi}{\partial x} = 0$$
$$\Phi = au + b\frac{\partial u}{\partial x}$$
Suppose I want to solve the above using ...
- 317
1
vote
1
answer
67
views
Computing material derivated of tensor quantity
I would like to compute the material derivated of a tensor quantity, in the context of the finite volume method (FVM):
The equation is:
$$
\frac{\mathrm{d} \textbf{T}}{\mathrm{d} t} = \frac{\partial \...
user42626
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