Browsing by Subject "Stabilization"

In this work, a stabilized finite element framework is developed to simulate small and large deformation solid mechanics problems involving complex geometries and complicated constitutive models. In particular, the focus is on solid dynamics problems involving nearly and fully incompressible materials. The work is divided into three main themes, the first is concerned with the development of stabilized finite element algorithms for hyperelastic materials, the second handles the case of viscoelastic materials, and the third focuses on algorithms for J2-plastic materials. For all three cases, problems in the small and large deformation regime are considered, and for the J2-plasticity case, both quasi-static and dynamic problems are examined.

Some of the key features of the algorithms developed in this work is the simplicity of their implementation into an existing finite element code, and their applicability to problems involving complicated geometries. The former is achieved by using a mixed formulation of the solid mechanics equations where the velocity and pressure unknowns are represented by linear shape functions, whereas the latter is realized by using triangular elements which offer numerous advantages compared to quadrilaterals, when meshing complicated geometries. To achieve the stability of the algorithm, a new approach is proposed in which the variational multiscale approach is applied to the mixed form of the solid mechanics equations written down as a first order system, whereby the pressure equation is cast in rate form.

Through a series of numerical simulations, it is shown that the stability properties of the proposed algorithm is invariant to the constitutive model and the time integrator used. By running convergence tests, the algorithm is shown to be second order accurate, in the $L^2$-nrom, for the displacements, velocities, and pressure. Finally, the robustness of the algorithm is showcased by considering realistic test cases involving complicated geometries and very large deformation.

Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.

The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.

Given a system's mathematical model, controller synthesis involves designing inputs to ensure desired responses. As models get more complicated to capture systems' behavior, so does synthesis. For instance, nonlinearities in differential equations invalidate linear design tools. Moreover, the state and input are typically constrained by physical limitations or safety concerns. Satisfying these constraints complicates the design. Further, since no model is perfect, the controller should account for the inevitable discrepancies. This thesis develops practical design tools by imposing specific structures on controllers and exploiting key structures in dynamical models.

The first part of this dissertation develops an offline, Lyapunov-based synthesis method for constrained control-affine systems that ensures stability and safety alongside secondary performance objectives. By limiting the search for the controller and Lyapunov function to continuous piecewise affine (CPA) functions, the Lyapunov conditions that must hold at all points in a compact set are turned into a non-convex program with a finite number of constraints defined on the vertices of a triangulated region. The non-convex program is then solved using a sequence of conservative, but well-posed semi-definite programs. This eliminates a priori design choices as well as taxing, non-convex optimizations common to existing methods. Two techniques to enlarge the region of attraction are discussed, including a state-dependent switched controller that further alleviates computations. Together with triangulation refinements, the method provides an efficient synthesis tool for constrained control-affine systems.

The second part focuses on systems modeled by a combination of subsystems. First, the previous part's method is adjusted to design for piecewise affine systems. For these systems, an important source of conservatism in design by CPA functions is eliminated, making the method competitive to those that rely on S-procedure or complex optimizations. Second, a method of constraint enforcement in presence of mode-dependent, additive disturbances for time-dependent switched systems is given. This approach extends an existing method that solves the problem for the deterministic case. The dwell-time characteristic of the external switching signal is exploited to design recursively feasible, switched tube-based model predictive controllers (MPCs). The online computations remain the same as that of a non-switched tube-based MPC.