Differential elimination for dynamical models via projections with applications to structural identifiability

Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available as a Julia package at https://github.com/SciML/StructuralIdentifiability.jl

Reaction of Beta Dicalcium-Silicate and Tricalcium-Silicate With Carbon-Dioxide and Water

148 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1978.Ope

Universal solutions in nonlinear anelasticity

In this work, we examine the existence and properties of universal solutions in nonlinear incompressible isotropic anelasticity. Universal solutions are those that exist for all members of a class of materials under the imposition of suitable boundary tractions. To this end, we provide a framework under which a wide array of different anelasticity theories can be recast, and use this framework to classify all anelastic universal solutions exhibiting particular symmetries, using known families of universal solutions in classical nonlinear elasticity as a starting point. We demonstrate that all known universal solutions possess one of these particular symmetries, prove that the classical universal solution families merge according to their symmetry groups once extended to the anelastic setting, and conjecture that such symmetries are necessary features of universal solutions, and hence that our classification is complete. In the process of doing this, we discover that two of these families not only possess generic solution branches depending on arbitrary functions, but also anomalous branches outside of these whose forms are fixed up to a finite number of constants. We provide graphical representations of examples from these anomalous branches, and discuss their possible applications.</p

The anelastic Ericksen problem: universal deformations and universal eigenstrains in incompressible nonlinear anelasticity

Ericksen’s problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate

Universal displacements in linear elasticity

In nonlinear elasticity, universal deformations are the deformations that exist for arbitrary strain-energy density functions and suitable tractions at the boundaries. Here, we discuss the equivalent problem for linear elasticity. We characterize the universal displacements of linear elasticity: those displacement fields that can be maintained by applying boundary tractions in the absence of body forces for any linear elastic solid in a given anisotropy class. We show that the universal displacements for compressible isotropic linear elastic solids are constant-divergence harmonic vector fields. We note that any divergence-free displacement field is a universal displacement for incompressible linear elastic solids. Further, we characterize the universal displacement fields for all the anisotropy classes, namely triclinic, monoclinic, tetragonal, trigonal, orthotropic, transversely isotropic, and cubic solids. As expected, universal displacements explicitly depend on the anisotropy class: the smaller the symmetry group, the smaller the space of universal displacements. In the extreme case of triclinic material where the symmetry group only contains the identity and minus identity, the only possible universal displacements are linear homogeneous functions

The mathematical foundations of anelasticity: Existence of smooth global intermediate configurations

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly

Differential elimination for dynamical models via projections with applications to structural identifiability

Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available as a Julia package at https://github.com/SciML/StructuralIdentifiability.jl