Questions tagged [tensor]
For questions involving computational modeling with tensors. The most common definitions of a tensor are a multilinear map or simply a multilinear array.
29
questions
1
vote
1answer
47 views
Converting a 4 rank matrix to 2 rank matrix after using tensorproduct
Let's say I have a 2x2 matrix (with symbols) called 'A'. Now, if do
B = sympy.tensorproduct(A,A)
print(sympy.shape(B))
I get,
...
1
vote
1answer
154 views
Confusion about bilinear form for elasticity equation in deal.ii tutorial
I'm learning how to solve vector-valued problems with deal.II library. In particular, I'm looking at the following introduction from the official website https://www.dealii.org/current/doxygen/deal.II/...
1
vote
0answers
80 views
3D Matrix (Tensor) Operating On a 1D Vector
Say you have a tensor $T$ and the components are represented by a 3 by 3 by 3 matrix. And you want to use that tensor to map a vector $u$ into a new vector $s$, both of which are 3 by 1 column vectors....
2
votes
0answers
33 views
Scaling tensor approximation by symmetric tensor decomposition with SciPy's L-BFGS-B
I am trying to approximate a symmetric tensor of which the values are in the range of [1e-7,1e-4], by a symmetric tensor decomposition of lower rank. For this I am using the L-BFGS-B method in SciPy's ...
5
votes
1answer
866 views
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 ...
1
vote
1answer
112 views
Gradient of dot product of two tensors
Through obtaining an alternative form for force balance equation in a fluid mechanics problem, I stopped at a point where I have to prove this identity where $A$ and $B$ are second-order matrices:$$\...
-1
votes
1answer
153 views
Numpys `tensordot` and what is happening mathematically
I've encountered a program where np.tensordot was used, so I tried looking it up but I can't really understand what this function is doing...
I feel rather ...
4
votes
1answer
70 views
Compute distances from a vector to a matrix of vectors
Let $\vec{a} \in \mathbb{R}^\alpha$, and let $H$ be a rank 3 tensor with dimensions $M_{[i \in \mathbb{N}]} \times N_{[j \in \mathbb{N}]} \times \alpha_{[k \in \mathbb{N}]}$ (where the subscript are ...
1
vote
0answers
37 views
Second fundamental form - Maple
I would like to know the command/line in Maple 16 or similar to obtain the second fundamental form tensor for a given metric. I've managed to obtain Rienmann and Ricci tensor, even Weyl, but I can't ...
4
votes
1answer
737 views
Is there a library that allows einstein summation on dense, sparse, and LinearOperator type tensors
Numpy's einsum only works with dense tensors.
Is there an alternative that also works with sparse tensors and linear operators?
For example, I might have a ...
1
vote
0answers
98 views
How to get the derivatives of the determinant and inverse of 2nd-order tensor wrt itself in SymPy?
I have a second-order tensor for which I need to compute the derivatives of its determinant and inverse w.r.t. itself. The equations are as follows:
$$\frac{\partial \, det(\mathbf{F})}{\partial F_{...
0
votes
1answer
252 views
Computing excited states using itensor (with DMRG)
I am trying to compute first few excited states of some Hamiltonian (I am using itensor and its DMRG algorithm). To do so, I am ...
0
votes
2answers
128 views
Second derivative in coordinate invariant form
To solve stationary, incompressible, inviscid and irrotational flow around a circular cylinder, I am using general coordinates. Since the flow is symmetrical, we only consider the upper half of the ...
1
vote
1answer
132 views
triple cross prouct of tensor
Im trying to compute a triple cross product of vectors a,b, and c in real space and integrate over the entire space. The result is a term in the hamiltonian for an electronic system so there are ...
1
vote
2answers
1k views
Derivative of the inverse of the Right Cauchy-Green Deformation Tensor wrt itself
In continuum mechanics, we define the Right-Cauchy-Green Deformation Tensor as
$\boldsymbol{C}=\boldsymbol{F}^T\boldsymbol{F}$
I want to compute $\frac{\partial \boldsymbol{C}^{-1}}{\partial \...
1
vote
0answers
97 views
Efficient out-of-place arbitrary rank GPU transpose
Summary: Is there an efficient out-of-place GPU tensor transpose operation that scales as $O(n)$ for tensors with $n$ total elements, regardless of the rank $d$? The naive algorithm costs $O(dn)$, ...
5
votes
3answers
2k views
Efficiently computing the product of a multi-dimensional matrix (or tensor) and vectors
Update: Thank you very much for all of you who answered below. I'm studying each answer now. In the long term, I'm more interested in solutions that work for sparse tensors (sorry I should have ...
1
vote
1answer
111 views
Binary tensor operations in Nutils [closed]
How does one write general tensor contractions in the Python-based finite element package Nutils? For example, how does one write the contraction of a fourth-order elasticity tensor $\boldsymbol{C}$ ...
9
votes
3answers
1k views
Second order tensor field visualization software
Is there an overview available over tensor visualization software?
My personal preference is:
A software which is free, well documented, and offers visualization techniques for different physical ...
0
votes
2answers
190 views
Vector and index notation equivalence
Given 2 vectors $\mathbf{u}$ and $\mathbf{v}$ the following are equivalent:
$\mathbf{u}\cdot\mathbf{v}$
$\mathbf{u}^T \mathbf{v}$
$u_i v_i$
$v_i u_i$
$\mathbf{v}\cdot\mathbf{u}$
$\mathbf{v}^T\...
1
vote
0answers
51 views
Optimization of nonlocal stencil-like operator on subset of regular grid
I am trying to optimize the execution time for this particular piece of fortran code.
Details:
i_gc is a (ngpts, 3) array of containing (i,j,k) indices for each grid point. This is a subset of the ...
0
votes
2answers
103 views
Explain this multivariate differential identity
$$ \frac{\partial|\nabla\phi|^2}{\partial\phi}=-2\nabla\cdot\nabla\phi$$
I would very appreciate that you help me . Please do it in detail, I am quite not good at such problems.
There is something ...
1
vote
0answers
185 views
What is the relation between Kruskal tensor and CP decomposition?
In Matlab Tensor Toolbox there is a tensor type called "Kruskal tensors", I found its form is similar to the CP decomposition.
Wikipedia mentioned:
"As such, many of the methods have been ...
1
vote
0answers
206 views
4th order tensor [closed]
I'm new with FEniCS and Python and I'm stuck with this issue:
is there a way to write a 4th order tensor in an easy way to implement?
I have to compute the following stiffnes tensor:
$A_{ijkl}= \...
1
vote
1answer
52 views
Anisotropic invariant expansion
I am trying to calculate the second and third invariants for a turbulent flow. I have the second order statistics (both transient and averaged). i.e $uu$, $vv$, $ww$, $uv$, $vw$ and $uw$. These are ...
1
vote
1answer
517 views
Any relation between the singular values of each flattening matrices and the core tensor out of Tucker decomposition?
Before I know how to do tucker decomposition, I mistakenly thought the core tensor is only from combining the singular value matrices of the flattening matrices. Yes I know it is not now.
For the ...
3
votes
1answer
238 views
Is there a reference/source paper for the TUCKER_ALS() in Tensor Toolbox for MATLAB?
TUCKER_ALS computes the best rank-(R1,R2,..,Rn) approximation of tensor X, according to the specified dimensions. I am using MATLAB Tensor Toolbox Version 2.5. I am wondering if I write a paper, how ...
2
votes
0answers
408 views
finite volume for diffusion equation with anisotropic (tensor) coefficient
Consider the scalar PDE for $u$ with Dirichlet boundary conditions:
$\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$,
$u = 0 \; \forall \; x\;\in \partial\Omega$
...
27
votes
10answers
18k views
Fast, lightweight C++ tensor library for dimension-agnostic code
I am looking for a C++ tensor library that supports dimension-agnostic code. Specifically, I need to perform operations along each dimension (up to 3), e.g. calculating a weighted sum. The dimensions ...
