An official website of the United States government
Abstract A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.
more »
« less
A Dual Variational Principle for Nonlinear Dislocation Dynamics
Abstract A dual variational principle is defined for the nonlinear system of PDE describing the dynamics of dislocations in elastic solids. The dual variational principle accounting for a specified set of initial and boundary conditions for a general class of PDE is also developed.
more »
« less
- Award ID(s):
- 2021019
- PAR ID:
- 10467916
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Elasticity
- ISSN:
- 0374-3535
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.more » « less
-
Abstract In this work, we investigate the existence and uniqueness properties of a composite structure (multilayered)–fluid interaction PDE system which arises in multi-physics problems, and particularly in biofluidic applications related to the mammalian blood transportation process. The PDE system under consideration consists of the interactive coupling of 3D Stokes flow and 3D elastic dynamics which gives rise to an additional 2D elastic equation on the boundary interface between these 3D PDE systems. By means of a nonstandard mixed variational formulation, we show that the PDE system generates a$$C_0$$ -semigroup on the associated finite energy space of data. In this work, the presence of the pressure term in the 3D Stokes equation adds a great challenge to our analysis. To overcome this difficulty, we follow a methodology which is based on the necessarily non-Leray-based elimination of the associated pressure term, via appropriate nonlocal operators. Moreover, while we express the fluid solution variable via decoupling of the Stokes equation, we construct the elastic solution variables by solving a mixed variational formulation via a Babuska–Brezzi approach.more » « less
-
Abstract We study a reaction–diffusion partial differential equation (PDE) system with a distributed input, subject to multiple unknown plant parameters with arbitrarily large uncertainties. Using Lyapunov-based techniques, we design a delay-adaptive predictor feedback controller that ensures local boundedness of system trajectories and asymptotic regulation of the closed-loop system in terms of the plant state. Specifically, we model the input delay as a one-dimensional transport PDE with a spatial variable, effectively transforming the time delay into a spatially distributed shift. For the resulting coupled transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping approach combined with the certainty-equivalence principle to derive an adaptive control law that compensates for both the unknown time delay and the unknown functional parameters. Simulation results are provided to illustrate the feasibility of our control design.more » « less
-
Abstract This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $$L_{2}$$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $$L_{2}$$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s10659-023-09998-5
