Amit Acharya's blog

Dynamic shear banding under adiabatic conditions in a mesoscale polycrystalline aggregate is studied using a model of mesoscale dislocation mechanics and experiments. The model involves a length scale related to hardening induced by excess/polar/geometrically necessary dislocation (GND) density, and utilizes a simple classical crystal plasticity model with isotropic Voce law hardening. Simulations of statistically representative volume elements of a polycrystal determined from experimental samples are conducted. Studies in 2-d (section) and 3-d capture the experimentally observed finite-width shear bands and the formation of low-angle subgrain boundaries even in the absence of heat conduction in the model, as well as size-dependent strengthening for grain sizes from 1 to 20μm. The 2-d and large-scale 3-d simulations, the latter involving 1 million finite elements, provide access to the progressive evolution of material strength, stress state, and temperature in the course of large deformations. GND distributions accumulate at grain boundaries and form patterned structures within grain interiors, offering insight into the microstructural changes that precede failure in adiabatic shear bands. Mesh-converged, delocalized and localized plastic flow to very large deformations without softening is observed for a significant range of parameters, reflecting a competition between GND hardening and thermal softening in setting the non-softening steady state in the absence of other ductile damage mechanisms in the model.

A model of dissipative micromagnetics coupled to (visco-)elasticity is explored, following the procedures of the Ericksen-Leslie theory of nematic liquid crystals allowing for angular momentum due to magnetization. An outcome is the Landau-Lifshitz-Gilbert theory coupled to material spin. A further power-less augmentation to the angular momentum of the theory with classical kinetic energy density is also considered, with a preliminary exploration of its potential in representing the Einstein-de Haas and Barnett effects within continuum mechanics. A treatment of the continuum mechanics of hard magnetic soft materials as a constrained polar material is presented. The models of DeSimone and James (2002) and Zhao et al. (2019) are discussed as two different, namely energetically and kinematically, constrained models of magnetoelasticity encompassed within the overall framework.

Sarthok. K. Baruah       Sabyasachi Chatterjee          Amit Acharya           Gerald J. Wang

The application of molecular dynamics (MD) simulations to quasi-static loading is severely limited by the large separation between atomic vibration timescales and experimentally relevant deformation rates. In this work, we employ the Practical Time Averaging (PTA) framework to overcome this limitation and enable atomistic simulations of crystalline solids under quasi-static loading conditions. PTA exploits the intrinsic separation of time scales by defining slow variables as time-averaged observables of the fast atomistic dynamics and their evolution in the slow loading timescale, thereby avoiding explicit integration of the fast dynamics. Using this approach, we simulate uniaxial deformation, in both tension and compression, of (4 to 20) nanometer sized cubic specimens of face-centered cubic Aluminum nanocrystals and applied strain rates approaching quasi-static conditions (10^−4 s−1 − 10^−3 s−1).

The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schr¨odinger equation, with the expected limiting form when the associated classical force system involves conservative forces.

Uditnarayan Kouskiya          Robert L. Pego        Amit Acharya

We look for traveling waves of the semi-discrete conservation law $4 \dot{u}_j + u_{j+1}^2 - u_{j-1}^2 = 0$, using variational principles related to concepts of ``hidden convexity'' appearing in recent studies of various PDE (partial differential equations). We analyze and numerically compute with two variational formulations related to dual convex optimization problems constrained by either the differential-difference equation (DDE) or nonlinear integral equation (NIE) that wave profiles should satisfy. We prove existence theorems conditional on the existence of extrema that satisfy a strict convexity criterion, and numerically exhibit a variety of localized, periodic and non-periodic wave phenomena.

Amit Acharya        Janusz Ginster

A scheme for generating a family of convex variational principles is developed, the Euler-Lagrange equations of each member of the family formally corresponding to the necessary conditions of optimal control of a given system of ordinary differential equations (ODE) in a well-defined sense. The scheme is applied to the Quadratic-Quadratic Regulator problem for which an explicit form of the functional is derived, and existence of minimizers of the variational principle is rigorously shown. It is shown that the Linear-Quadratic Regulator problem with time-dependent forcing can be solved within the formalism without requiring any nonlinear considerations, in contrast to the use of a Riccati system in the classical methodology.

Our work demonstrates a pathway for solving nonlinear control problems via convex optimization.

Dissipative models for the quasi-static and dynamic response due to slip in an elastic body containing a single slip plane of vanishing thickness are developed. Discrete dislocations with continuously distributed cores can glide on this plane, and the models are developed as special cases of a fully three-dimensional theory of plasticity induced by dislocation motion. The reduced models are compared and contrasted with the augmented Peierls model of dislocation dynamics. A primary distinguishing feature of the reduced models is the a-priori accounting of space-time conservation of Burgers vector during dislocation evolution. A physical shortcoming of the developed models as well as the Peierls model with regard to a dependence on the choice of a distinguished, coherent reference configuration is discussed, and a testable model without such dependence is also proposed.

A scheme for treating the Second Law of thermodynamics as a constraint and accounting for the approximate nature of constitutive assumptions in continuum thermomechanics is discussed. An unconstrained, concave, variational principle is designed for solving the resulting mathematical problem. Cases when the Second Law becomes an over-constraint on the mechanical model, as well as when it serves as a necessary constraint, are discussed.

Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants

N. Sukumar              Amit Acharya

Many partial differential equations (PDEs) such as Navier–Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the L2 norm and H1 seminorm for the steady-state convection-diffusion problem.