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4th order tensor rotation - sources to refer

I am trying to model a linear elastic material in Abaqus using a UMAT. For my application, I need to rotate the 6x6 compliance matrix for a given set of eigenvectors (or a rotation matrix). I came across a thread titled "debugging a rotation matrix…
Sagar Shah
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Numerical Green functions for a nonlinear wave equation

I am trying to put down some code to get numerically the solution of the following PDE: $$ \partial^2_t\phi-\partial^2_x\phi+\lambda\phi^3=\delta(x)\delta(t). $$ Of course, there are several problems already starting with a proper numerical…
Jon
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Finding points inside cells of power (generalized Voronoi) diagram

Suppose we have a set of points $p_1,\ldots,p_n\in\mathbb R^d$ as well as a set of weights $w_1,\ldots,w_n\in\mathbb R$. Recall that the power cell associated to the pair $(p_k,w_k)$ is given by: $$\mathcal C_k:=\{x\in\mathbb…
Justin Solomon
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FEM toolbox for discretization of higher order PDEs

Is there any (open source) FEM toolbox that allows the direct discretization of higher order PDEs without the need to split them up into systems of second order?
Thomas W.
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Numerical flux and source term in FVM (Burger's like equation)

I'm trying to solve the following equation with FVM $$u_t + f(u)_x = g(u)$$ where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except for the source term. My main problem is that I don't…
VoB
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Numerically stable and fast sum of last K elements in sequence

Suppose I have a long, possibly infinite, sequence $x := [x_1, x_2, ...]$, and I want to use it to compute another sequence $y:=[y_1, y_2, ...]$ where each element is the sum of the last K elements of the input sequence. i.e. $y_i = \sum_{j=max(1,…
Peter
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Estimate extreme eigenvalues with CG

CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest eigenvalue is generally well approximated whereas the…
Tom
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Fast computation of the zeros of a trigonometric polynomial

I am wondering what is the fastest way to compute the zeroes of the trigonometric polynomial $$ T(x) = \sum_{i=1}^La_i\sin\big(2\pi(a_i x) - \phi_i \big), \\ a_i \in \mathbb{N}, \\ \phi_i \in \left[-\pi,\pi\right[, \\ x \in …
Arrigo
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evaluating $\coth(x) - 1/x$ for real $x$, on 2 "pieces"

The function $\coth(x) - 1/x$ has a removable singularity at 0. Its Taylor series is: $$ \coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots $$ I would like to evaluate the first 3 terms of the Taylor series for $|x| \le…
Glenn Davis
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Best hardware solution for microsecond Molecular Dynamics

We would like to reach Molecular Dynamics simulation of proteins with around 20000 atoms in explicit water with trajectories of around 1 microsecond each. We are looking at different options for computer resources to complete these…
Open the way
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Block-matrix: optimal fill-in reduction for LU factorization

Consider a square $N \times N$ block-matrix $\mathbf{A}$, where each $n \times n$ block $\mathbf{A}_{ii}$ is either a dense block or a zero-block. So, $N$ denotes the number of blocks, $n$ denotes the size of each square block. Random example for…
Anton Menshov
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Is a complete bacteria simulation with an exascale supercomputer possible?

Will it be possible to simulate a complete (at least simple) bacteria atom by atom on an exascale supercomputer? or is it possible already today with the largest systems? Here, I've read that scientists can simulate an organelle of bacteria. For…
Open Food Broker
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fastest way to compute many small dot products

I have two n-by-3 blocks contiguous in memory ("n vectors of length 3") and I'd like to compute the dot product between each of the rows as fast as possible. In numpy, using einsum is the fastest variant to the best of my knowledge…
Nico Schlömer
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How to check if my stiffness matrix is correct

I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "irregular" shape. The mesh has been given to me by…
slamWolfen
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Understanding the Eisenstat-Walker method for choosing the tolerance of a linear solver when solving a non-linear PDE

We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we are interested in optimally choosing the…
BlaB
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