Questions tagged [boundary-conditions]
For questions regarding the choice and/or appropriateness of conditions necessary to model a particular phenomenon with a partial differential equations.
408
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Symmetry axis boundary condition
I was wondering about the symmetry axis boundary condition in commercial CFD solvers such as ANSYS Fluent.
If the problem is the flow through a round pipe or out of a round nozzle, it is natural to ...
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0
answers
55
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How to set Neumann BC for coupled transport problem in weak form?
Consider
$$\begin{aligned}
\partial_t v + b\cdot \nabla \phi &=0 \\
\partial_t \phi + b\cdot \nabla v &= 0
\end{aligned}$$
for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
2
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2
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160
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Starting configuration for Molecular Dynamics
I have to build a bilayer using a structure obtained in a previous MD simulation. In the structure, some chains of the molecules cross the simulation box as shown below. I added water layers to the ...
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2
votes
1
answer
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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 ...
- 384
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3
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173
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Dirichlet condition in finite element method
I'm trying to understand the approach described in these questions:
How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices
How to apply non zero ...
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3
votes
1
answer
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First derivative central differences with reflecting boundary conditions
I have the following problem in 1-D:
\begin{equation}
\partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega.
\end{...
- 1,271
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2
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How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?
I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
3
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1
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311
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Solving Poisson's Equation with Periodic Boundary Conditions
So, I've been attempting to design a simple solver for a problem of finding the gravitational potential of a system using Poisson's equation (let's call the potential phi, $\phi$). The goal is that I ...
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1
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1
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147
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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 ...
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1
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How can we use the two boundary conditions for the Taylor-Maccoll Equations for Cone Shock Waves?
The Taylor-Maccoll equations below govern the gas dynamics around a supersonic, axially oriented cone.
Using the notation from Anderson's Fundamentals of Aerodynamics, which uses a dimensionless ...
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0
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Why do we use modified pressure in incompressible multiphase solvers with gravity?
The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
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1
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49
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Why is the maximum potential energy greater than the maximum kinetic energy?
I was plotting the energy variation in a mass-spring system. If I define the initial conditions to be at maximum displacement from the origin, the potential energy is plotted correctly but kinetic ...
- 123
0
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0
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82
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Analytical Equation of the gaussian 1D wave equation with periodic Boundary condition
I am trying to validate the 1D analytical wave equation with a numerical solution with periodic boundary conditions. I have implemented the periodic boundary condition for the numerically calculated ...
- 121
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0
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87
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How to include zero flux boundary conditions?
I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme:
For $0< \theta \leq \pi$
\begin{align}
\rho_i^{n+1}=\rho_i^{n}+D\...
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0
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111
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Interface condition for 1D Helmholtz equation using finite element method
I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is;
$$c^2\nabla^2p + w^2p=0 \qquad \...
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0
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boundary condition at rotation axis of a spherically-symmetric system
The quantity I am interested in is not the rotation rate $\Omega$, but I will use this quantity nonetheless to make the problem clearer. I am interested in a spherically-symmetric system and in the ...
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0
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Why do BVP solvers' APIs only allow "unknown" parameters in the derivative and residual functions but not "known" parameters?
I recently needed to solve a second order boundary value problem and noticed that both scipy.integrate.solve_bvp and Matlab's ...
- 465
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0
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93
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Compact Finite Differences for the Heat Equation with Robin Boundary Conditions
I am trying to solve the Heat equation with Robin Boundary condition:
$$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$
for $ 0\leq x\leq1$ ...
- 21
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1
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Nonlinear Robin boundary condition involving square root
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
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Want to know how to solve this high order differential equation system with boundary condition?
The numerical results may be like this:
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1
answer
133
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Free Time Dependent Schrodinger Equation with Inhomogeneous Dirichlet boundary
There exists a FFT-based method to solve the poisson equation in inhomogeneous Dirichlet boundary condition using the sine-transform. For example, Which fourier series is needed to solve a 2D poisson ...
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2
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138
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Practical implementation of the discrete compatibility condtion
I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form
$$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
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2
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0
answers
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Boundary conditions for compressible Euler equations
I want to solve the compressible 1D Euler equations numerically.
Theory says that for subsonic inflows, one can prescribe two variables, e.g. pressure and temperature. Density can then be computed ...
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0
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943
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What EXACTLY are the "outflow" and "pressure outlet" BCs in ANSYS Fluent?
I am just starting my MSc and we have been given an assignment in which we must model two-dimensional, laminar, incompressible Poiseuille flow in a channel. We are supposed to use a VELOCITY INLET ...
- 51
1
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0
answers
69
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How to incorporate a continuity of 1st derivatives at an internal interface in a 2nd order Poisson equation in finite elements method?
How should I modify the following matrix formulation of a finite element (FE) approximation of Poisson equation at the (internal) interface denoted by the nodes $i-1$ to $i+1$ having a continuous 1st ...
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1
vote
1
answer
37
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Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions
I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
- 153
2
votes
0
answers
108
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Numerical precision on tricky coupled nonlinear boundary value problem on infinite interval
I am trying to solve with high precision the following coupled system $(f,h)$ on $[0,\infty]$:
$$-h''-\frac{1}{r}h'+\lambda_c h(f^2-1)+4\lambda_h h^3=0$$
$$-f''-\frac{1}{r}f'+\frac{1}{r^2}f+ f(-\...
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1
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154
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Boundary value problem solver fails on trivial case
I am trying to solve a boundary value problem on $[0, \infty]$, using scipy's scipy.integrate.solve_bvp and I am seeing that the solutions are not converging even ...
- 33
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1
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164
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Can successive over-relaxation (SOR) method deal with ill-posed PDE BVP?
Recently, I've been struggling to understand the limitations and capabilities of the successive over-relaxation (SOR) method for boundary value problems which are ill-posed, such as, for instance, ...
- 53
1
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1
answer
233
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Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)
I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form:
$$\nabla(\epsilon\nabla\varphi)=\nabla\...
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4
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1
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243
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Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
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How can I define an equipotential surface/volume in FEniCS?
I want to solve electrostatic problem for potential. Charge density and medium permittivity are known, so is the potential of a grounded surface. I know how I can implement that.
But I would like to ...
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0
answers
32
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How does Tannehill impose boundary conditions when coding the Parabolized Navier Stokes on an Implicit Finite Differences Scheme?
I'm trying to implement the scheme he describes on his book "Computational Fluid Mechanics and Heat Transfer" on Chap.9 and I'm having trouble imposing BC.
I don’t get how he imposes them. I ...
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1
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Applying Stress Boundary Conditions in Commercial Finite Element Analysis Codes
I am trying to replicate a finite element analysis given in a research paper titled On the Detection of Stress Singularities in Finite Element Analysis 1 by G.B.Sinclair et. al.
The geometry of the ...
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1
vote
0
answers
43
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Adding wall friction to a 3D model - Stokes equation
Maybe that is not the best place for the following question, if such, let me know and sorry.
I would like to model the flow of expanding bentonite in a fracture following the given reference. In order ...
- 99
0
votes
0
answers
544
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The mathematical meaning of a zero gradient pressure boundary condition in the Navier-Stokes equations
I would like to solve the Navier-Stokes equations for the unsteady problem of the flow around a circular cylinder. I would like to understand how to write mathematically the boundary condition for the ...
0
votes
2
answers
291
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Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?
I'm solving a local instability problem for a pipe Poiseuille flow.
The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial).
The basic flow is $\bar{u_r}=0, \bar{u_\...
- 1
1
vote
0
answers
230
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Open boundary condition for 1d wave equation with variable wave speed using finite differences
I have implemented a finite difference solver for the 1d wave equation with variable wave speed:
$$ u_{tt} = c(x)u_{xx}, \hspace{10mm}c(x) = \dfrac{6 -x^2}{2} \hspace{5mm} $$
on $-2 \leq x \leq 2, t &...
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1
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924
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Who can explain the minimum image convention in molecular dynamic simulations?
How to choose the cutoff radius so that the atoms do not interact with its periodic image? Especially when simulating macromolecules (proteins).
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66
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Passing boundary conditions to solver
Quite broad question,
Currently building own Poisson solver subroutine for CFD solver.
Works smooth, the goal is to generalise the input and make it flexible.
Description of also:
Memory allocation.
...
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votes
3
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131
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Simple to program method for elliptic PDE with curved boundary?
I know very little about numerical PDE, so I apologize if this equation is ill-formed (and I appreciate any corrections).
I am currently learning about Brownian motion. A classic result is that we can ...
- 105
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0
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79
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Transparent Boundary Conditions for Finite Difference ADI PR 2D TDSE solution
I want to put (non-dirichlet) boundary conditions inside the code I wrote to solve the 2dim TDSE using the alternating direction implicit Peaceman - Rachford method.
$$
(1 + iB\Delta t/2 ) \psi^{n+1/2}...
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3
votes
0
answers
66
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Change in Variables applied to biharmonic equation
Background
I want to solve the following biharmonic equation:
$$\frac{ \partial^4 s }{ {\partial \xi}^4 }+\frac{ \partial^4 s }{ {\partial \xi}^2{\partial t}^2 }+\frac{ \partial^4 s }{ {\partial t}^4 }...
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2
votes
1
answer
621
views
Solving ODE with Spectral Method using Chebyshev Polynomials
I would like to solve following the basic equation of linear elasticity (for simplicity in 1D)
$$
\frac{d}{dx} \left( E \frac{du}{dx} \right) = 0 \quad \textrm{with} \quad u(1)=0, \; u(-1)=b
$$
...
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1
vote
1
answer
88
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How to Impose nonhomogeneous Neumann Boundary Condition in the DG Formulation
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
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2
votes
1
answer
172
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DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
- 83
1
vote
1
answer
44
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An explanation of 2delta waves on non-staggered grids
While looking into the difference between staggered and collocated grids, I came across an effect called $2\Delta x$-oscillations, which happen on non-staggered grids, but not on staggered grids. This ...
- 105
2
votes
0
answers
121
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switching boundary conditions to simulate a reciprocating flow in COMSOL
This is a re-posted question which was poorly defined earlier
I have been trying to simulate reciprocating flow of a convective fluid through a heated channel in COMSOL-Multiphysics. The schematic of ...
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3
votes
2
answers
409
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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 $\...
- 281
3
votes
0
answers
52
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Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?
I am trying to simulate the following equation using FDTD
$ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $
subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
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