Spandan Maiti

Recycling Krylov Subspaces for Sequences of Linear Systems

Many problems in science and engineering require the solution of a long sequence of slowly changing linear systems. We propose and analyze two methods that significantly reduce the total number of matrix-vector products required to solve all systems. We consider the general case where both the matrix and right-hand side change, and we make no assumptions regarding the change in the right-hand sides. Furthermore, we consider general nonsingular matrices, and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix. Our methods work well under these general assumptions, and hence form a significant advancement with respect to related work in this area. We can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems. We consider two approaches that allow for the continuous improvement of the recycled subspace at low cost. We consider both Hermitian and non-Hermitian problems, and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling. We also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics, materials science, and computational physics.

In vivo study of magnesium plate and screw degradation and bone fracture healing.

Each year, millions of Americans suffer bone fractures, often requiring internal fixation. Current devices, like plates and screws, are made with permanent metals or resorbable polymers. Permanent metals provide strength and biocompatibility, but cause long-term complications and may require removal. Resorbable polymers reduce long-term complications, but are unsuitable for many load-bearing applications. To mitigate complications, degradable magnesium (Mg) alloys are being developed for craniofacial and orthopedic applications. Their combination of strength and degradation make them ideal for bone fixation. Previously, we conducted a pilot study comparing Mg and titanium devices with a rabbit ulna fracture model. We observed Mg device degradation, with uninhibited healing. Interestingly, we observed bone formation around degrading Mg, but not titanium, devices. These results highlighted the potential for these fixation devices. To better assess their efficacy, we conducted a more thorough study assessing 99.9% Mg devices in a similar rabbit ulna fracture model. Device degradation, fracture healing, and bone formation were evaluated using microcomputed tomography, histology and biomechanical tests. We observed device degradation throughout, and calculated a corrosion rate of 0.40±0.04mm/year after 8 weeks. In addition, we observed fracture healing by 8 weeks, and maturation after 16 weeks. In accordance with our pilot study, we observed bone formation surrounding Mg devices, with complete overgrowth by 16 weeks. Bend tests revealed no difference in flexural load of healed ulnae with Mg devices compared to intact ulnae. These data suggest that Mg devices provide stabilization to facilitate healing, while degrading and stimulating new bone formation.

Magnesium alloys as a biomaterial for degradable craniofacial screws.

Recently, magnesium (Mg) alloys have received significant attention as potential biomaterials for degradable implants, and this study was directed at evaluating the suitability of Mg for craniofacial bone screws. The objective was to implant screws fabricated from commercially available pure Mg and alloy AZ31 in vivo in a rabbit mandible. First, Mg and AZ31 screws were compared to stainless steel screws in an in vitro pull-out test and determined to have a similar holding strength (∼40N). A finite-element model of the screw was created using the pull-out test data, and this model can be used for future Mg alloy screw design. Then, Mg and AZ31 screws were implanted for 4, 8 and 12weeks, with two controls of an osteotomy site (hole) with no implant and a stainless steel screw implanted for 12weeks. Microcomputed tomography was used to assess bone remodeling and Mg/AZ31 degradation, both visually and qualitatively through volume fraction measurements for all time points. Histological analysis was also completed for the Mg and AZ31 at 12weeks. The results showed that craniofacial bone remodeling occurred around both Mg and AZ31 screws. Pure Mg had a different degradation profile than AZ31; however, bone growth occurred around both screw types. The degradation rate of both Mg and AZ31 screws in the bone marrow space and the muscle were faster than in the cortical bone space at 12weeks. Furthermore, it was shown that by alloying Mg, the degradation profile could be changed. These results indicate the promise of using Mg alloys for craniofacial applications.

Continuum and Molecular-Level Modeling of Fatigue Crack Retardation in Self-Healing Polymers

A numerical model to study the fatigue crack retardation in a self-healing material (White et al., 2001, Nature, 409, pp. 794-797) is presented. The approach relies on a combination of cohesive modeling for fatigue crack propagation and a contact algorithm to enforce crack closure due to an artificial wedge in the wake of the crack. The healing kinetics of the self-healing material is captured by introducing along the fracture plane a state variable representing the evolving degree of cure of the healing agent. The atomic-scale processes during the cure of the healing agent are modeled using a coarse-grain molecular dynamics model specifically developed for this purpose. This approach yields the cure kinetics and the mechanical properties as a function of the degree of cure, information that is transmitted to the continuum-scale models. The incorporation of healing kinetics in the model enables us to study the competition between fatigue crack growth and crack retardation mechanisms in this new class of materials. A systematic study of the effect of different loading and healing parameters shows a good qualitative agreement between experimental observations and simulation results.

Force production and mechanical accommodation during convergent extension

Forces generated within the embryo during convergent extension (CE) must overcome mechanical resistance to push the head away from the rear. As mechanical resistance increases more than eightfold during CE and can vary twofold from individual to individual, we have proposed that developmental programs must include mechanical accommodation in order to maintain robust morphogenesis. To test this idea and investigate the processes that generate forces within early embryos, we developed a novel gel-based sensor to report force production as a tissue changes shape; we find that the mean stress produced by CE is 5.0±1.6 Pascal (Pa). Experiments with the gel-based force sensor resulted in three findings. (1) Force production and mechanical resistance can be coupled through myosin contractility. The coupling of these processes can be hidden unless affected tissues are challenged by physical constraints. (2) CE is mechanically adaptive; dorsal tissues can increase force production up to threefold to overcome a stiffer microenvironment. These findings demonstrate that mechanical accommodation can ensure robust morphogenetic movements against environmental and genetic variation that might otherwise perturb development and growth. (3) Force production is distributed between neural and mesodermal tissues in the dorsal isolate, and the notochord, a central structure involved in patterning vertebrate morphogenesis, is not required for force production during late gastrulation and early neurulation. Our findings suggest that genetic factors that coordinately alter force production and mechanical resistance are common during morphogenesis, and that their cryptic roles can be revealed when tissues are challenged by controlled biophysical constraints. Highlighted article: A new method for measuring tissue-scale force production is used to investigate the mechanical control of convergent extension in Xenopus embryos.

Computationally efficient black-box modeling for feasibility analysis

Computational cost is a major issue in modern large-scale simulations used across different disciplines of science and engineering. Computationally efficient surrogate models that can represent the original model with desired accuracy have been explored in the recent past. However, with the exception of few efforts, most of these techniques rely on a reduced order representation of the original complex model, resulting in a loss of information. In this paper we demonstrate the applicability of high dimensional model representation (HDMR) technique in addressing this issue while preserving the original model dimension. We will discuss the applicability of this surrogate modeling technique in the field of feasibility analysis drawing examples from process systems and materials design. It will be shown that the original physical models can be essentially considered as a black box, and same methodology can be applied across all the examples studied. It is found that the accuracy of the surrogate models depends on the order of the approximation and number of sampling points employed. While first-order approximation is largely inadequate, second-order approximation is sufficient for the model systems studied. Sampling requirement is also dramatically low for the construction of these surrogate models.

Local heating and viscosity drop during shear band evolution in bulk metallic glasses under quasistatic loading

Based on the facts that the thickness of a shear band in bulk metallic glasses (BMGs) is a few tens of nanometers, the shear displacement across the band is few micrometers, and the time for their formation is in submicrosecond duration, the local strain rates within the shear band can be as high as 109∕s. To capture such dynamic effects, a thermo-micromechanical model based on momentum diffusion mechanism, free-volume theory, and heat diffusion analysis is proposed. The model has been shown to capture the characteristic rate effects, i.e., significant local temperature rise and a dramatic drop in viscosity during shear band evolution in BMGs. The model also takes into account the effects of normal stress component on the deformation behavior of BMGs. While the predicted maximum temperature rise under quasistatic deformation in the absence of normal component of stress is low (300 K), significant temperature rise well above 1000 K accompanied by a sudden drop in viscosity has been predicted under dynamic ...

A mechanistic model on the role of “radially-running” collagen fibers on dissection properties of human ascending thoracic aorta.

Aortic dissection (AoD) is a common condition that often leads to life-threatening cardiovascular emergency. From a biomechanics viewpoint, AoD involves failure of load-bearing microstructural components of the aortic wall, mainly elastin and collagen fibers. Delamination strength of the aortic wall depends on the load-bearing capacity and local micro-architecture of these fibers, which may vary with age, disease and aortic location. Therefore, quantifying the role of fiber micro-architecture on the delamination strength of the aortic wall may lead to improved understanding of AoD. We present an experimentally-driven modeling paradigm towards this goal. Specifically, we utilize collagen fiber micro-architecture, obtained in a parallel study from multi-photon microscopy, in a predictive mechanistic framework to characterize the delamination strength. We then validate our model against peel test experiments on human aortic strips and utilize the model to predict the delamination strength of separate aortic strips and compare with experimental findings. We observe that the number density and failure energy of the radially-running collagen fibers control the peel strength. Furthermore, our model suggests that the lower delamination strength previously found for the circumferential direction in human aorta is related to a lower number density of radially-running collagen fibers in that direction. Our model sets the stage for an expanded future study that could predict AoD propagation in patient-specific aortic geometries and better understand factors that may influence propensity for occurrence.

The Impact of Boundary Conditions on Surface Curvature of Polypropylene Mesh in Response to Uniaxial Loading

Exposure following pelvic organ prolapse repair has been observationally associated with wrinkling of the implanted mesh. The purpose of this study was to quantify the impact of variable boundary conditions on the out-of-plane deformations of mesh subjected to tensile loading. Using photogrammetry and surface curvature analyses, deformed geometries were accessed for two commercially available products. Relative to standard clamping methods, the amount of out-of-plane deformation significantly increased when point loads were introduced to simulate suture fixation in-vivo. These data support the hypothesis that regional increases in the concentration of mesh potentially enhance the host׳s foreign body response, leading to exposure.

Varying degrees of nonlinear mechanical behavior arising from geometric differences of urogynecological meshes

Synthetic polypropylene meshes were designed to restore pelvic organ support for women suffering from pelvic organ prolapse; however, the FDA released two notifications regarding potential complications associated with mesh implantation. Our aim was to characterize the structural properties of Restorelle and UltraPro subjected to uniaxial tension along perpendicular directions, and then model the tensile behavior of these meshes utilizing a co-rotational finite element model, with an imbedded linear or fiber-recruitment local stress-strain relationship. Both meshes exhibited a highly nonlinear stress-strain behavior; Restorelle had no significant differences between the two perpendicular directions, while UltraPro had a 93% difference in the low (initial) stiffness (p=0.009) between loading directions. Our model predicted that early alignment of the mesh segments in the loading direction and subsequent stretching could explain the observed nonlinear tensile behavior. However, a nonlinear stress-strain response in the stretching regime, that may be inherent to the mesh segment, was required to better capture experimental results. Utilizing a nonlinear fiber recruitment model with two parameters A and B, we observed improved agreement between the simulations and the experimental results. An inverse analysis found A=120 MPa and B=1.75 for Restorelle (RMSE=0.36). This approach yielded A=30 MPa and B=3.5 for UltraPro along one direction (RMSE=0.652), while the perpendicular orientation resulted in A=130 MPa and B=4.75 (RMSE=4.36). From the uniaxial protocol, Restorelle was found to have little variance in structural properties along these two perpendicular directions; however, UltraPro was found to behave anisotropically.