Abstract
Tensors.jl is a Julia package that provides efficient computations with symmetric and non-symmetric tensors. The focus is on the kind of tensors commonly used in e.g. continuum mechanics and fluid dynamics. Exploiting Julia’s ability to overload Unicode infix operators and using Unicode in identifiers, implemented tensor expressions commonly look very similar to their mathematical writing. This possibly reduces the number of bugs in implementations. Operations on tensors are often compiled into the minimum assembly instructions required, and, when beneficial, SIMD-instructions are used. Computations involving symmetric tensors take symmetry into account to reduce computational cost. Automatic differentiation is supported, which means that most functions written in pure Julia can be efficiently differentiated without having to implement the derivative by hand. The package is useful in applications where efficient tensor operations are required, e.g. in the Finite Element Method. Funding statement: Support for this research was provided by the Swedish Research Council (VR), grant no. 621-2013-3901 and grant no. 2015-05422.
Highlights
Partial Differential Equations (PDEs) describing natural phenomena are modelled using tensors of different order.Two commonly studied problems are heat transfer, which include temperature and heat flux, and continuum mechanics, which include stress and strain and the so-called tangent stiffness
For a deformation gradient F = I + ∇ ⊗ u, where u is the displacement from the reference to the current configuration, the right Cauchy-Green deformation tensor is defined by C = F T · F
We have presented the Julia package Tensors.jl
Summary
Tensors.jl is a Julia package that provides efficient computations with symmetric and non-symmetric tensors. Exploiting Julia’s ability to overload Unicode infix operators and using Unicode in identifiers, implemented tensor expressions commonly look very similar to their mathematical writing. This possibly reduces the number of bugs in implementations. Operations on tensors are often compiled into the minimum assembly instructions required, and, when beneficial, SIMD-instructions are used. Automatic differentiation is supported, which means that most functions written in pure Julia can be efficiently differentiated without having to implement the derivative by hand. The package is useful in applications where efficient tensor operations are required, e.g. in the Finite Element Method
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