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Register a new userStochastic modelling of symmetric positive-definite material tensors
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Added by
Sharana Kumar Shivanand
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tensors where one wants to be able to prescribe certain classes of spatial symmetries and invariances for the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly higher spatial invariance class. Our special interest is in the class of physically symmetric and positive definite tensors, as they appear often in the description of materials. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear vector space of physically symmetric tensors, it may be advantageous to widen the notion of mean to the so-called Frechet mean, which is based on distance measures between positive definite tensors other than the usual Euclidean one. For the sake of simplicity, as well as to expose the main idea as clearly as possible, we limit ourselves here to second order tensors. It is shown how the random ensemble can be modelled and generated, with fine control of the spatial symmetry or invariance of the whole ensemble, as well as its Frechet mean, independently in its scaling and directional aspects. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is explored. It is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest, such as temperature distribution and heat flux.
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We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations. Instead of directly predicting the stress, the SPD-NN trains a neural network to predict the Cholesky factor of a tangent stiffness matrix, based on which the stress is calculated in the incremental form. As a result of the special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of the SPD-NN make it a promising tool for a wide range of constitutive modeling applications.
In this paper, a new class of positive semi-definite tensors, the MO tensor, is introduced. It is inspired by the structure of Moler matrix, a class of test matrices. Then we focus on two special cases in the MO-tensors: Sup-MO tensor and essential MO tensor. They are proved to be positive definite tensors. Especially, the smallest H-eigenvalue of a Sup-MO tensor is positive and tends to zero as the dimension tends to infinity, and an essential MO tensor is also a completely positive tensor.
In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. In the DPG context, the primal problem is simply the system of linear equations coming from a standard DPG method and the dual problem is a similar system of equations, coming from a new method which is dual to DPG. This new method has the same coefficient matrix as the associated DPG method but has a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poissons boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of the goal-oriented adaptive mesh refinement strategies for quantities of interest with either interior or boundary terms.
We present a class of reduced basis (RB) methods for the iterative solution of parametrized symmetric positive-definite (SPD) linear systems. The essential ingredients are a Galerkin projection of the underlying parametrized system onto a reduced basis space to obtain a reduced system; an adaptive greedy algorithm to efficiently determine sampling parameters and associated basis vectors; an offline-online computational procedure and a multi-fidelity approach to decouple the construction and application phases of the reduced basis method; and solution procedures to employ the reduced basis approximation as a {em stand-alone iterative solver} or as a {em preconditioner} in the conjugate gradient method. We present numerical examples to demonstrate the performance of the proposed methods in comparison with multigrid methods. Numerical results show that, when applied to solve linear systems resulting from discretizing the Poissons equations, the speed of convergence of our methods matches or surpasses that of the multigrid-preconditioned conjugate gradient method, while their computational cost per iteration is significantly smaller providing a feasible alternative when the multigrid approach is out of reach due to timing or memory constraints for large systems. Moreover, numerical results verify that this new class of reduced basis methods, when applied as a stand-alone solver or as a preconditioner, is capable of achieving the accuracy at the level of the {em truth approximation} which is far beyond the RB level.
In an iterative approach for solving linear systems with ill-conditioned, symmetric positive definite (SPD) kernel matrices, both fast matrix-vector products and fast preconditioning operations are required. Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an $mathcal{H}^2$ representation or an equivalent fast multipole method representation. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the $mathcal{H}^2$ representation, such as the hierarchically semiseparable (HSS) matrix representation, which provides fast solve operations. Previously, an algorithm was presented to construct an HSS approximation to an SPD kernel matrix that is guaranteed to be SPD. However, this algorithm has quadratic cost and was only designed for recursive binary partitionings of the points defining the kernel matrix. This paper presents a general algorithm for constructing an SPD HSS approximation. Importantly, the algorithm uses the $mathcal{H}^2$ representation of the SPD matrix to reduce its computational complexity from quadratic to quasilinear. Numerical experiments illustrate how this SPD HSS approximation performs as a preconditioner for solving linear systems arising from a range of kernel functions.
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