Questions tagged [variational-calculus]
For questions about modeling the change of functionals with respect to input functions. This could include how to solve a particular functional derivative or finding the function that minimizes a given functional.
12
questions
5
votes
1
answer
265
views
Does the weighted residual method not use energy minimization in any form?
I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
- 73
2
votes
0
answers
160
views
Best way to compute given functional with accuracy:
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1},
$$
Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
- 119
2
votes
2
answers
414
views
Weak form of the Navier-Cauchy equation
I am trying to obtain the weak form of the Navier-Cauchy equation, which is
$$- \rho \omega ^2 \textbf{U} - \mu \nabla ^2 \textbf{U} - (\mu + \lambda) \nabla (\nabla \cdot \textbf{U}) = \textbf{F}$$
...
- 115
1
vote
1
answer
86
views
Prove that the set of maximizers are independent of parameter in the objective function
A maximization problem reads as
$$ J(y) = \sum_{k=1}^{K} \sigma_k(y)^q \mathop{\rightarrow}^{y} max$$
where $q \in [1,\infty]$ is a user-defined parameter and functions $\sigma_k, k=\{1,\dots,K\}$ ...
- 95
7
votes
2
answers
2k
views
How to solve calculus of variations problems numerically?
For example, how to solve the well-known isoperimetric problem (i.e., to enclose the largest area with a fixed-length curve)?
We can simplify things a bit and fix the two ends of the curve at $[a,0]$,...
- 277
0
votes
1
answer
94
views
Deriving weak form of a set of scalar equations
I have the equilibrium equation in elasticity for a static case.i.e
Div T=0.
For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is ...
1
vote
0
answers
117
views
functional second derivative
I'm trying to build a numerical solution for a parameter estimation problem of reaction-diffusion equation, using the adjoint method.
To summarize it, I'm trying to minimize the function
$$ G=\frac{...
- 139
1
vote
0
answers
205
views
General-purpose Numerical Calculus of Variations Problem
Calculus of variations problems are generally cast in in the following simple form: find $u(t)$ that satisfies some boundary conditions and minimises
$$
F[u] = \int_{t=0}^{t=t_f} f(u(t),u'(t),t) dt.
$...
- 231
3
votes
1
answer
188
views
Numerical computation of the velocity in the steady Navier-Stokes equation
I've asked this question on Math.SE too.
Let
$d\in\left\{1,\ldots,4\right\}$
$\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open and $\partial\Lambda$ be Lipschitz
$V:=\left\{u\in H_0^1(\...
- 283
3
votes
0
answers
206
views
Solving a nonlinear poisson equation via variational minimization
I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem.
$- \nabla ((1 + u^2) \nabla u) = f$
$u = 0 \ \text{on} \ \Omega $
I am using Newton solver, where I have ...
- 31
2
votes
1
answer
525
views
Finite element method applied to variational problem/functional VS weak formulation
I am confused.
I read an introduction to finite element method where it was derived for the poisson equation:
$$-\Delta u + cu = f,\qquad, u = g_0 \text{ on Dirichlet boundaries},\qquad\partial_n u ...
- 123
3
votes
0
answers
157
views
Reference Request: Variational Problem
I want to solve approximately the following variational problem:
Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and $\...
- 131