Geometric elasticity for graphics, simulation, and computation

We develop new algorithms which combine the rigorous theory of mathematical elasticity with the geometric underpinnings and computational attractiveness of modern tools in geometry processing. We develop a simple elastic energy based on the Biot strain measure, which improves on state-of-the-art methods in geometry processing. We use this energy within a constrained optimization problem to, for the first time, provide surface parameterization tools which guarantee injectivity and bounded distortion, are user-directable, and which scale to large meshes. With the help of some new generalizations in the computation of matrix functions and their derivative, we extend our methods to a large class of hyperelastic stored energy functions quadratic in piecewise analytic strain measures, including the Hencky (logarithmic) strain, opening up a wide range of possibilities for robust and efficient nonlinear elastic simulation and geometry processing by elastic analogy.

On the solution of first excursion problems by simulation with applications to probabilistic seismic performance assessment

In a probabilistic assessment of the performance of structures subjected to uncertain environmental loads such as earthquakes, an important problem is to determine the probability that the structural response exceeds some specified limits within a given duration of interest. This problem is known as the first excursion problem, and it has been a challenging problem in the theory of stochastic dynamics and reliability analysis. In spite of the enormous amount of attention the problem has received, there is no procedure available for its general solution, especially for engineering problems of interest where the complexity of the system is large and the failure probability is small.

The application of simulation methods to solving the first excursion problem is investigated in this dissertation, with the objective of assessing the probabilistic performance of structures subjected to uncertain earthquake excitations modeled by stochastic processes. From a simulation perspective, the major difficulty in the first excursion problem comes from the large number of uncertain parameters often encountered in the stochastic description of the excitation. Existing simulation tools are examined, with special regard to their applicability in problems with a large number of uncertain parameters. Two efficient simulation methods are developed to solve the first excursion problem. The first method is developed specifically for linear dynamical systems, and it is found to be extremely efficient compared to existing techniques. The second method is more robust to the type of problem, and it is applicable to general dynamical systems. It is efficient for estimating small failure probabilities because the computational effort grows at a much slower rate with decreasing failure probability than standard Monte Carlo simulation. The simulation methods are applied to assess the probabilistic performance of structures subjected to uncertain earthquake excitation. Failure analysis is also carried out using the samples generated during simulation, which provide insight into the probable scenarios that will occur given that a structure fails.

Temporal characteristic simulation for stimulated emission depletion microscopy

The imaging technology of stimulated emission depletion (STED) utilizes the nonlinearity relationship between the fluorescence saturation and the excited state stimulated depletion. It implements three-dimensional (3D) imaging and breaks the diffraction barrier of far-field light microscopy by restricting fluorescent molecules at a sub-diffraction spot. In order to improve the resolution which attained by this technology, the computer simulation on temporal behavior of population probabilities of the sample was made in this paper, and the optimized parameters such as intensity, duration and delay time of the STED pulse were given.

Extracting material response from simple mechanical tests on hardening-softening-hardening viscoplastic solids

Compliant foams are usually characterized by a wide range of desirable mechanical properties. These properties include viscoelasticity at different temperatures, energy absorption, recoverability under cyclic loading, impact resistance, and thermal, electrical, acoustic and radiation-resistance. Some foams contain nano-sized features and are used in small-scale devices. This implies that the characteristic dimensions of foams span multiple length scales, rendering modeling their mechanical properties difficult. Continuum mechanics-based models capture some salient experimental features like the linear elastic regime, followed by non-linear plateau stress regime. However, they lack mesostructural physical details. This makes them incapable of accurately predicting local peaks in stress and strain distributions, which significantly affect the deformation paths. Atomistic methods are capable of capturing the physical origins of deformation at smaller scales, but suffer from impractical computational intensity. Capturing deformation at the so-called meso-scale, which is capable of describing the phenomenon at a continuum level, but with some physical insights, requires developing new theoretical approaches.

A fundamental question that motivates the modeling of foams is ‘how to extract the intrinsic material response from simple mechanical test data, such as stress vs. strain response?’ A 3D model was developed to simulate the mechanical response of foam-type materials. The novelty of this model includes unique features such as the hardening-softening-hardening material response, strain rate-dependence, and plastically compressible solids with plastic non-normality. Suggestive links from atomistic simulations of foams were borrowed to formulate a physically informed hardening material input function. Motivated by a model that qualitatively captured the response of foam-type vertically aligned carbon nanotube (VACNT) pillars under uniaxial compression [2011,“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech.Phys. Solids, 59, pp. 2227–2237, Erratum 60, 1753–1756 (2012)], the property space exploration was advanced to three types of simple mechanical tests: 1) uniaxial compression, 2) uniaxial tension, and 3) nanoindentation with a conical and a flat-punch tip. The simulations attempt to explain some of the salient features in experimental data, like
1) The initial linear elastic response.
2) One or more nonlinear instabilities, yielding, and hardening.

The model-inherent relationships between the material properties and the overall stress-strain behavior were validated against the available experimental data. The material properties include the gradient in stiffness along the height, plastic and elastic compressibility, and hardening. Each of these tests was evaluated in terms of their efficiency in extracting material properties. The uniaxial simulation results proved to be a combination of structural and material influences. Out of all deformation paths, flat-punch indentation proved to be superior since it is the most sensitive in capturing the material properties.

Dynamic rupture simulation integrated with earthquake observations

Dynamic rupture simulations are unique in their contributions to the study of earthquake physics. The current rapid development of dynamic rupture simulations poses several new questions: Do the simulations reflect the real world? Do the simulations have predictive power? Which one should we believe when the simulations disagree? This thesis illustrates how integration with observations can help address these questions and reduce the effects of non-uniqueness of both dynamic rupture simulations and kinematic inversion problems. Dynamic rupture simulations with observational constraints can effectively identify non-physical features inferred from observations. Moreover, the integrative technique can also provide more physical insights into the mechanisms of earthquakes. This thesis demonstrates two examples of such kinds of integration: dynamic rupture simulations of the M_w 9.0 2011 Tohoku-Oki earthquake and of earthquake ruptures in damaged fault zones:

(1) We develop simulations of the Tohoku-Oki earthquake based on a variety of observations and minimum assumptions of model parameters. The simulations provide realistic estimations of stress drop and fracture energy of the region and explain the physical mechanisms of high-frequency radiation in the deep region. We also find that the overridding subduction wedge contributes significantly to the up-dip rupture propagation and large final slip in the shallow region. Such findings are also applicable to other megathrust earthquakes.

(2) Damaged fault zones are usually found around natural faults, but their effects on earthquake ruptures have been largely unknown. We simulate earthquake ruptures in damaged fault zones with material properties constrained by seismic and geological observations. We show that reflected waves in fault zones are effective at generating pulse-like ruptures and head waves tend to accelerate and decelerate rupture speeds. These mechanisms are robust in natural fault zones with large attenuation and off-fault plasticity. Moreover, earthquakes in damaged fault zones can propagate at super-Rayleigh speeds that are unstable in homogeneous media. Supershear transitions in fault zones do not require large fault stresses. In the end, we present observations in the Big Bear region, where variability of rupture speeds of small earthquakes correlates with the laterally variable materials in a damaged fault zone.

Direct simulation of quantum dynamics in complex systems

Accurate simulation of quantum dynamics in complex systems poses a fundamental theoretical challenge with immediate application to problems in biological catalysis, charge transfer, and solar energy conversion. The varied length- and timescales that characterize these kinds of processes necessitate development of novel simulation methodology that can both accurately evolve the coupled quantum and classical degrees of freedom and also be easily applicable to large, complex systems. In the following dissertation, the problems of quantum dynamics in complex systems are explored through direct simulation using path-integral methods as well as application of state-of-the-art analytical rate theories.

Acceleration of ultra-thin electron layer.Analytical treatment compared with 1D-PIC simulation

In this paper, we apply an analytical model [V.V. Kulagin et al., Phys. Plasmas 14, 113101 (2007)] to describe the acceleration of an ultra-thin electron layer by a schematic single-cycle laser pulse and compare with one-dimensional particle-in-cell (1D-PIC) simulations. This is in the context of creating a relativistic mirror for coherent backscattering and supplements two related papers in this EPJD volume. The model is shown to reproduce the 1D-PIC results almost quantitatively for the short time of a few laser periods sufficient for the backscattering of ultra-short probe pulses.