Probabilistic finite elements

In the Probabilistic Finite Element Method (PFEM), finite element methods have been efficiently combined with second-order perturbation techniques to provide an effective method for informing the designer of the range of response which is likely in a given problem. The designer must provide as input the statistical character of the input variables, such as yield strength, load magnitude, and Young's modulus, by specifying their mean values and their variances. The output then consists of the mean response and the variance in the response. Thus the designer is given a much broader picture of the predicted performance than with simply a single response curve. These methods are applicable to a wide class of problems, provided that the scale of randomness is not too large and the probabilistic density functions possess decaying tails. By incorporating the computational techniques we have developed in the past 3 years for efficiency, the probabilistic finite element methods are capable of handling large systems with many sources of uncertainties. Sample results for an elastic-plastic ten-bar structure and an elastic-plastic plane continuum with a circular hole subject to cyclic loadings with the yield stress on the random field are given

Probabilistic finite elements for fatigue and fracture analysis

An overview of the probabilistic finite element method (PFEM) developed by the authors and their colleagues in recent years is presented. The primary focus is placed on the development of PFEM for both structural mechanics problems and fracture mechanics problems. The perturbation techniques are used as major tools for the analytical derivation. The following topics are covered: (1) representation and discretization of random fields; (2) development of PFEM for the general linear transient problem and nonlinear elasticity using Hu-Washizu variational principle; (3) computational aspects; (4) discussions of the application of PFEM to the reliability analysis of both brittle fracture and fatigue; and (5) a stochastic computational tool based on stochastic boundary element (SBEM). Results are obtained for the reliability index and corresponding probability of failure for: (1) fatigue crack growth; (2) defect geometry; (3) fatigue parameters; and (4) applied loads. These results show that initial defect is a critical parameter

Development of finite element procedures for fluid-structure interaction

In this thesis the development of finite element procedures for fluid-structure interaction problems is presented. The areas upon which attention is focused are: numerical transient algorithms which emphasize implicit-explicit finite element concepts; finite element kinematical descriptions for modelling fluid subdomains in fluidstructure interaction problems; finite element methodology for nearly incompressible fluids and solids, and beam, plate and shell structures based upon theories which include transverse shear deformations; and finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. All these nonlinear methodologies have been integrated into a working finite element computer code. A number of numerical examples are presented to demonstrate the effectiveness of these approaches

Mesoscale simulation of stress relaxation in thin polymer films and the connection to nanocomposites

Key insight into interphase formation and confinement effects in nanocomposites has recently come from studies on polymer thin films supported on solid substrates. In these thin films, both the free surface and the solid supporting layer cause complex changes in the behavior of the polymer. The range and magnitude of these effects have been singled out by systematically varying the boundary conditions (free standing film, supported thin film, and polymer layer confined between two surfaces) and surface/polymer chemistry. Most importantly, the Schadler group and the Torkelson group have shown a quantitative equivalence between nanocomposites and thin films with regards to glass-transition temperature (Tg) via the calculation of an equivalent metric of confinement within the nanocomposite from the distribution of filler surface-to-surface distances. This finding is important because it allows for direct prediction of the Tg of the nanocomposite directly from thin film measurements and microstructural statistics, leveraging current capabilities in accurate computational/experimental characterization of film properties. However, it is currently unknown whether the thin-film analogy can be extended into the constitutive behavior of polymer nanocomposites, most importantly the stress relaxation behavior of the matrix that governs viscoelastic behavior. With an ultimate aim to address this issue, we have begun examining the stress-relaxation in doubly supported polymer thin films through coarse grained simulation using the FENE model. The current study elucidates the connection among film thickness, interfacial energy, and stress relaxation dynamics. In order to characterize the dynamic relaxation behavior of the films at constant temperature, we calculate via an extended, tensorial Green–Kubo relation the linear shear-relaxation modulus from equilibrium coarse-grained simulations of the bulk and of films of varying thickness. We then compare the simulated relaxation moduli to both the Rouse model and the theory of Likhtman and McLeish (originally based on the based on the tube model), with the additional changes proposed by Hou, Svaneborg, Everaers, and Grest. Applications to the continuum mechanics of both thin films and nanocomposites will be discussed

Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly-multivariable system with incomplete governing equations. In this study, we embed the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant and unique dimensionless numbers and scaling laws from data. The proposed methodology, called dimensionless learning, can reduce high-dimensional parametric spaces into descriptions involving just a few physically-interpretable dimensionless parameters, which significantly simplifies the process design and optimization of the system. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. The examples include turbulent Rayleigh-Benard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. We also show that the proposed approach can identify dimensionally-homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques

Endocytosis of PEGylated nanoparticles accompanied by structural and free energy changes of the grafted polyethylene glycol

AbstractNanoparticles (NPs) are in use to efficiently deliver drug molecules into diseased cells. The surfaces of NPs are usually grafted with polyethylene glycol (PEG) polymers, during so-called PEGylation, to improve water solubility, avoid aggregation, and prevent opsonization during blood circulation. The interplay between grafting density σp and grafted PEG polymerization degree N makes cellular uptake of PEGylated NPs distinct from that of bare NPs. To understand the role played by grafted PEG polymers, we study the endocytosis of 8 nm sized PEGylated NPs with different σp and N through large scale dissipative particle dynamics (DPD) simulations. The free energy change Fpolymer of grafted PEG polymers, before and after endocytosis, is identified to have an effect which is comparable to, or even larger than, the bending energy of the membrane during endocytosis. Based on self-consistent field theory Fpolymer is found to be dependent on both σp and N. By incorporating Fpolymer, the critical ligand-receptor binding strength for PEGylated NPs to be internalized can be correctly predicted by a simple analytical equation. Without considering Fpolymer, it turns out impossible to predict whether the PEGylated NPs will be delivered into the diseased cells. These simulation results and theoretical analysis not only provide new insights into the endocytosis process of PEGylated NPs, but also shed light on the underlying physical mechanisms, which can be utilized for designing efficient PEGylated NP-based therapeutic carriers with improved cellular targeting and uptake

Coarse-grained simulation of recovery in thermally activated shape-memory polymers

Thermally actuated shape-memory polymers (SMPs) are capable of being programmed into a temporary shape and then recovering their permanent reference shape upon exposure to heat, which facilitates a phase transition that allows dramatic increase in molecular mobility. Experimental, analytical, and computational studies have established empirical relations of the thermomechanical behavior of SMPs that have been instrumental in device design. However, the underlying mechanisms of the recovery behavior and dependence on polymer microstructure remain to be fully understood for copolymer systems. This presents an opportunity for bottom–up studies through molecular modeling; however, the limited time-scales of atomistic simulations prohibit the study of key performance metrics pertaining to recovery. In order to elucidate the effects of phase fraction, recovery temperature, and deformation temperature on shape recovery, here we investigate the shape-memory behavior in a copolymer model with coarse-grained potentials using a two-phase molecular model that reproduces physical crosslinking. Our simulation protocol allows observation of upwards of 90% strain recovery in some cases, at timescales that are on the order of the timescale of the relevant relaxation mechanism (stress relaxation in the unentangled soft phase). Partial disintegration of the glassy phase during mechanical deformation is found to contribute to irrecoverable strain. Temperature dependence of the recovery indicates nearly full elastic recovery above the trigger temperature, which is near the glass-transition temperature of the rubbery switching matrix. We find that the trigger temperature is also directly correlated with the deformation temperature, indicating that deformation temperature influences the recovery temperatures required to obtain a given amount of shape recovery, until the plateau regions overlap above the transition region. Increasing the fraction of glassy phase results in higher strain recovery at low to intermediate temperatures, a widening of the transition region, and an eventual crossover at high temperatures. Our results corroborate experimental findings on shape-memory behavior and provide new insight into factors governing deformation recovery that can be leveraged in biomaterials design. The established computational methodology can be extended in straightforward ways to investigate the effects of monomer -chemistry, low--molecular-weight solvents, physical and chemical crosslinking, different phase--separation morphologies, and more complicated mechanical deformation toward predictive modeling capabilities for stimuli-responsive -polymers

Nanomedicine

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Microtubule-driven conformational changes in platelet -morphogenesis

The influence of the range of the involved geometric and material parameters, such as the available area for conformational changes, the bilayer thickness, the interaction energy between transmembrane domains and lipids, is largely explored. Bounds on the available conformations experienced by the transmebrane domains are also provided

Microelastic wave field signatures and their implications for microstructure identification

AbstractThis work combines closed-form and computational analyses to elucidate the dynamic properties, termed signatures, of waves propagating through solids defined by the theory of elasticity with microstructure and the potential of such properties to identify microstructure evolution over a material’s lifetime. First, the study presents analytical dispersion relations and frequency-dependent velocities of waves propagating in microelastic solids. A detailed parametric analysis of the results show that elastic solids with microstructure recover traditional gradient elasticity under certain conditions but demonstrate a higher degree of flexibility in adapting to observed wave forms across a wide frequency spectrum. In addition, a set of simulations demonstrates the ability of the model to quantify the presence of damage, just another type of microstructure, through fitting of the model parameters, especially the one associated with the characteristic length scale of the underlying microstructure, to an explicit geometric representation of voids in different damage states