Burt Tilley and Vadim Yakovlev receive $775,336 award from AFOSR titled ``Permittivity Gradients, Polarization, and Gas Dynamics in Composite Electromagnetic Heat Exchangers’
In Professor Tilley's and Professor Yakovlev's own words, the project consists of a five-year research program to quantify the fundamental heat-transfer processes in the conversion efficiency from incoming electromagnetic radiation into elevated internal energy of a compressible coolant. The idea is to use electromagnetic-radiation absorbing materials, either porous or designed with channels through which a coolant can flow, that can withstand temperature up to 2000 K, heat these materials through the application of electromagnetic waves, and then run coolant through the material to harness the desired energy. Since electrical conductivity of these materials depends on temperature, multiple steady temperatures are possible at the same input power. The research program centers on using asymptotic multiscale methods including homogenization to formulate an effective medium theory to describe the energy conservation and electric field propagation through this medium, for incompressible and compressible coolants. The spatial variation of loss factor will be examined in order to compensate for conduction losses found at higher temperatures, and as a simple model for metal-ceramic composites. These results depend on the examination of the energy transfer between the composite and the coolant, and we shall consider these modes for incompressible and compressible coolants. Three-dimensional electric field amplitude equations, developed in our current award using high-frequency homogenization, will be extended to incorporate heat transfer, viscous fluid flow spatial dimensions, and comparisons with solutions using finite-difference time-domain and with finite-element methods will be performed. The goal is to develop a systematic method to better understand wave propagation, heat transfer and power delivery to the coolant for a general three-dimensional spatially-periodic microstructure. Learn more about
Luca Capogna Receives Five-Year Grant to Use Mathematics to Classify and Understand the Nature of Shapes
Luca Capogna, professor and head of the Department of Mathematical Sciences, has received a five-year, $42,000 grant from the Simons Foundation, a 24-year-old organization that funds research into mathematics and the sciences. The project being funded is titled “Topics in Nonlinear PDE and in Quasiconformal Mappings.”
The goal of Capogna’s research is to classify all possible shapes—whether the curls of a lettuce leaf shape or the space swept up by a robot arm as it moves through all of its possible configurations—using a mix of geometry and analysis methods. By better classifying and understanding shapes, scientists will better understand their nature and how they behave.
The research should enable Capogna to determine whether or not two spaces are similar. For instance, two shapes may look very different but have hidden similarities, from a mathematical perspective.
Capogna’s project, which is a continuation of research he’s been conducting for the past 10 years, could ultimately be applied to the study of cloud computing or even social networks. For instance, he is using mathematical models that relate to curved spaces, such as a curling lettuce leaf, to understand spaces that expand very quickly, a technique that connects a leaf that takes up more space as it uncurls to a cloud of data or a social network that rapidly expand in size.
Award Date: July 31, 2018
William Martin Receives $150,000 NSF Grant to Further Our Understanding of Association Schemes
William Martin, professor of mathematical sciences, has received a three-year, $150,000 grant from the National Science Foundation for a research project titled, “Association Schemes and Configurations in Real and Complex Space.”
The project is aimed at taking on questions that have stumped the best mathematical minds in the world for decades. Martin is looking to advance mathematicians’ understanding of association schemes, which are finite combinatorial structures that can be viewed algebraically, geometrically, or as highly symmetric networks.
Finding new association schemes, or even better understanding them, could enable researchers to use one quantum computer to simulate another. They also could be used to discover new error-correcting codes for digital communications, new spherical designs for estimating solutions to calculus problems in high-dimensional space, or new secure scrambling components for symmetric key encryption.
Martin is working on underlying algebraic theorems that will lead to a stronger mathematical theory, though he said he is focused on making advances that other researchers can then use in their own work, to solve problems in various applied areas.
At least one PhD student and multiple MQP teams will be working on the project.
Award Date: Aug. 2, 2018
Zheyang Wu Receives NSF Award for Work Related to ALS
Zheyang Wu, associate professor of mathematical sciences, has received a three-year, $150,000 award from the National Science Foundation for a project titles, “Optimal and Adaptive p-Value Combination Methods with Application to ALS Exome Sequencing Study.” The project is focused on developing tests that will give scientists a better understanding of which DNA segments are related to ALS susceptibility. Using ALS exome-sequence data, researchers will develop better data analysis methodology to paint a clearer picture of how genes influence ALS.
ALS, or amyotrophic lateral sclerosis, is a progressive neurodegenerative disease that affects nerve cells in the brain and spinal cord, affecting one’s speech and ability to swallow and to control the muscles. Ultimately, it causes paralysis and death. Each year, more than 5,000 people in the United States are diagnosed with ALS, which is also known as Lou Gehrig's disease.
Genetics plays a critical role in ALS. Despite numerous advances in recent years, doctors still cannot trace the genetic cause of a significant amount of ALS cases. This research project is taking on the “missing heritability” problem, using innovative genetic data analysis algorithms and more powerful p-value combination tests to analyze large exome sequencing data for detecting novel ALS genes. The methodology being developed in this project has broad applications and could be used to better understand other diseases, as well. A WPI graduate student will work on this research with Wu.
Andrea Arnold Receives NSF Award to Develop Computational Filtering Methods
August 16, 2018
Andrea Arnold, assistant professor of mathematical sciences, has receives a three-year, $220,458 award from the National Science Foundation for a project titled “Computational Filtering Methods for Time-Varying Parameter Estimation in Nonlinear Systems. ” The aim of the study is to design and analyze novel computational methods for estimating time-varying parameters through use of nonlinear filtering. The research has applications in the life sciences, such as modeling the spread of infectious diseases, determining the optimal treatment strategy for HIV drug therapy, and modeling tissue response to laser-based microsurgery.
Arnold’s research in applied mathematics focuses on inverse problems and uncertainty quantification, which involves estimating unknown system parameters using indirect observations and analyzing the changes in predicted outcomes due to changes in the inputs. Many applications in modern science involve system parameters that are estimated using little prior information. This poses a challenge in applied and computational mathematics, particularly for problems where knowledge of parameters is crucial in obtaining trustworthy model output.
The study will develop mathematically sound and computationally efficient systematic approaches for estimating time-varying parameters with unknown dynamics. This work also will involve developing models for parameter evolution that take into account any prior knowledge relating to the structure or behavior of the parameter over time without defining explicit functions to describe the dynamics. Arnold will include WPI undergraduates and graduate students on the project team starting in the summer of 2019.
Bayesian Benchmarking of County Estimates for Agricultural Commodities Under Inequality constraints
Research and Development Division (RDD) at the National Agricultural Statistics Service (NASS), of the agencies of the USDA
The research involved builds on Professor Nandram's prior experience
working with NASS on Objective Yield Survey and Interview Yield Survey for
forecasting US corn yields. In professor Nandram's own description of the
work: When the county estimates are obtained, they are benchmarked to
district and state level estimates that are more reliable.
The next step is to incorporate inequality constraints into the Bayesian
benchmarking procedure. When Farm Service Agency of the USDA provides earlier estimates of acreage,
and the county estimates provided by NASS must be larger than the FSA estimates such that the these
estimates must add up to the target (state estimate). This is a very
challenging small-area problem as the feasible region is
Tight. Some commodities we work on are corn, soybean, wheat. These
estimates are useful for conservation and allocation of disaster relief.
Adaptive Resource Management Enabling Deception (ARMED), Extreme DDoS (Distributed Denial of Service) Defense (XD3)
Raytheon BBN Technologies
The ARMED project is developing techniques to protect service enclaves from extreme distributed denial of service (DDOS) attacks. These attacks may abuse flaws and ill-specified aspects of the network stack running at the service's endpoint. Usually, these attacks are not detected until it is too late, and the core mathematical tools leveraged in this project include robust techniques for linear and non-linear dimension reduction. Under this award, Dr. Paffenroth will investigate the concept of augmenting the usual protocol processing with additional data collection, analysis, and response capabilities, with a focus on anomaly detection.
New Mathematical Methods for Fracture Evolution
National Science Foundation
This research project concerns fundamental mathematical questions in fracture mechanics, an area of importance in materials and structural engineering. Despite substantial recent progress in mathematical analysis of models for fracture and crack propagation, nucleation and propagation of material defects in general, and fracture in particular, remain poorly understood, yet their accurate prediction is of great importance in many materials science applications. This project aims to develop new mathematical methods for addressing some of the major challenges in this area. These include showing existence of solutions to classes of mathematical models for fracture evolution, improving dynamic fracture models, and analyzing properties of dynamic fracture solutions, with a particular emphasis on exploring crack branching and its consequences.
Showing existence of quasi-static cohesive fracture evolutions, showing existence for mathematical models of dynamic fracture, and establishing qualitative properties of dynamic fracture solutions are major challenges in the mathematical analysis of fracture mechanics. The methods that have been used to show existence for quasi-static Griffith evolutions are now known to fail for cohesive fracture. The main difficulty arises from the delicate role that history plays in the definition of these solutions. This project will continue the development of new methods for analyzing this and other quasi-static problems, based on higher order energy approximations using history at only a finite number of prior times. Dynamic Griffith fracture is also very delicate, due to complex interactions between elastic singularities and the (a priori unknown) evolving crack set. New methods based on blow-up techniques will be developed for analyzing these evolutions. Learn more about the award.
Higher-Order Methods for Interface Problems with Non-Aligned Meshes
National Science Foundation
Interface problems arise in several applications including heart models, cochlea models, aquatic animal locomotion, blood cell motion, front-tracking in porous media flows and material science, to name a few. One of the difficulties in these problems is that solutions are normally not smooth across interfaces, and therefore standard numerical methods will lose accuracy near the interface unless the meshes align to it. However, it is advantageous to have meshes that do not align with the interface, especially for time dependent problems where the interface moves with time. Re-meshing at every time step can be prohibitively costly, can destroy the structure of the grid, can deteriorate the well-conditioning of the stiffness matrix, and affect the stability of the problem. The first problem studied will involve new stable and higher-order accurate Finite Element - Immersed Boundary Methods (FE-IBM) for evolution problems where the interface moves with time. The second problem studied is the design and analysis of robust higher-order discretizations for interface problems with high-contrast discontinuous diffusion coefficients. Benefits of the project include the strengthening of connections between numerical analysis and other areas of science and engineering, particularly bioengineering, porous media flows, material sciences and parallel computing. This project will impact the development of numerical algorithms used in the fluid-structure interaction communities. A broader impact will be the training of graduate and undergraduate students of mathematics and related disciplines by exposing them to interdisciplinary problems and collaborations addressing questions of great technological importance. One of the drawbacks of the finite element and finite difference immersed boundary methods is that they are only first-order accurate due to the non-smoothness of the solution across the interface. Furthermore, very few mathematical analyses of these methods exist for time dependent problems and for fluid-structure interaction problems. The first part of the project involves the construction of higher-order FE-IBM algorithms and establishing a corresponding mathematical foundation to obtain rigorous time stability and a priori and a posteriori error estimates. The second part of the project deals with new finite element methods which are able to accurately capture solutions of elliptic interface problems with high-contrast coefficients in the case that the finite element mesh is not necessarily aligned with the interface. The goal here is to develop finite element methods with optimal convergence rates, where the constants hidden in these estimates are independent of the contrast and on how the mesh crosses the interface. Learn more about the grant.
Multiscale Methods in Beamed Energy Harnessing Applications
Air Force Office of Scientific Research (AFOSR)
We are interested in understanding how electromagnetic-radiation absorbing materials, either porous or designed with channels through which a coolant can flow, can be used to transfer the energy from electromagnetic radiation to a coolant. For the applications of interest, temperatures can reach 2000 K, and since the electrical conductivity of these materials depends on temperature, multiple steady temperatures are possible at the same input power. The research program centers on using asymptotic multi-scale methods including homogenization to formulate an effective-medium theory to describe the energy conservation and electric field amplitude propagation through this medium, for incompressible and compressible coolants. Results will be compared to GPU-accelerated finite-difference time-domain (FDTD) scripts in two spatial dimensions. Simulations in three dimensions will be done by implementing computational approaches from the 2D-FDTD schemes within a three-dimensional spatial framework in COMSOL Multi-physics.
Small Area Estimation at USDA's National Agricultural Statistics Service Research and Development Division (RDD) at the National Agricultural Statistics Service (NASS)
This grant builds on Professor Nandram's prior experience working with NASS on Objective Yield Survey and Interview Yield Survey for forecasting US corn yields. As part of the new grant, Professor Nandram will participate in the new NASS small area estimation research program. In this program he will continue his work in applications of Bayesian small area estimation to government programs such as the National Health and Nutrition Examination Survey and the National Health Interview Survey to assess the health of the US population. The Research and Development Division (RDD) at the National Agricultural Statistics Service (NASS) employs a number of recent PhDs and researchers working on their dissertations. As part of the grant, Professor Nandram will also provide research mentoring for these individuals.
3D MRI-Based Modeling for Computer-Aided Right Ventricle Remodeling Surgery
NIH/NHLBI, 1 R01 HL089269
Tang, D. WPI PI; Harvard PIs: Pedro del Nido (contact), Tal Geva)
Right ventricular dysfunction developing late after congenital cardiac surgery is one of the most common causes of heart failure in adults with congenital heart disease. Current surgical management of late RV dysfunction, consisting of pulmonary valve insertion and reduction of the RV outflow patch, reduces RV volume but does not result in a predictable improvement in RV function. In this project, we propose to develop a computational modeling approach to determine the efficacy and suitability of the various reconstructive options to treat failing RV in ToF pts. We will use non-invasive cardiac magnetic resonance imaging (CMR) to provide patient-specific RV/LV morphology, deformation, and flow data for the construction and validation of computational models. 3D CMR-based RV/LV combination models will be constructed, which include fluid-structure interactions (RV/LV and RV patch), two-layer RV/LV structure, anisotropic material properties, fiber orientation, and active contraction to simulate blood flow, heart motion, and stress/strain distribution to evaluate the effect of different remodeling procedures on RV function, and to seek an optimal RV volume and patch design to improve post-operative RV function. Clinical imaging and hemodynamic data from an ongoing NHLBI-funded clinical trial will be used to build and validate the model. Our ultimate goal is to apply this methodology in patient- specific computer-aided cardiac surgery planning to reach optimal surgical procedure design and outcome in patients with RV dysfunction from congenital heart defects. Learn more about the grant.
CAREER: Numerical Methods and Biomechanical Models for Sperm Motility
National Science Foundation
Mammalian sperm must navigate the female reproductive tract, swimming a distance greater than 1000 times their own length to reach and fertilize the egg. In order to aid in the treatment of reduced sperm motility, it is important to understand interactions of the sperm flagellum with different regions of the reproductive tract. In particular, fluid flow helps bring the egg to the uterus (in the opposite direction of sperm progression). Recent experiments have shown that a large percentage of sperm exhibit positive rheotaxis, the ability to reorient and swim against a background flow. Additionally, sperm will bind and unbind to the oviductal wall and the role of a background flow on sperm detachment is not known. The main scientific goals of this project include further analyzing existing experimental data (through image processing techniques) and developing new computational models to understand the clinical importance of migration through the female reproductive tract and sperm binding and detachment from walls in a background flow. Several new computational modeling frameworks will be developed to allow simulations of sperm in the presence of a background flow and a wall. The PI will provide interdisciplinary training for several students (undergraduate and graduate) as well as one postdoc in the areas of computational biofluids, image processing of experimental movies, and model development. In addition, the PI will work to develop image processing and modeling modules to be used in area High Schools and at summer programs for High School students at WPI. Learn more about the grant.
Topics in quasiconformal mappings and subelliptic PDE
National Science Foundation
SubRiemannian geometry and subelliptic partial differential equations (PDEs) are used to model real life systems where there is a constrained dynamics. Examples of such systems include the motion of robot arms, structural functions of the first layer of the mammalian visual cortex, the Black-Scholes model for financial markets and quantum computing. Geometric and analytic properties of such spaces are captured in a quantitative fashion by studying the behavior of certain families of transformations of the space into itself. This project aims at studying fine properties of such transformations. In particular, the proposed research will provide a theoretical basis for implementing numerical simulations of real-life system. Learn more about the grant.
TWC: Small: Towards Practical Fully Homomorphic Encryption
National Science Foundation
WPI's Vernam Lab (formerly the CRIS Lab) has been involved in the practical implementation of cryptographic algorithms, as well as their security analysis, for over a dozen years. The current project is a continuation of their work on implementation issues in homomorphic encryption. With Berk Sunar (ECE) as PI and Bill Martin (MA/CS) as coPI, the team aims to assess and adapt recently proposed cryptographic primitives for applications where homomorphic properties are Required. An encryption algorithm is additively homomorphic if it allows a third party to efficiently compute an encryption of, say, x+y given only encryptions of x and y and neither the values themselves nor the decryption key. Am encryption scheme is "fully homomorphic" (hence called an "FHE" scheme) when it permits arbitrary computations on ciphertexts without compromising security. Solving a 30-year-old open problem, Gentry proposed the first FHE scheme in 2009 and, since then, a diverse assortment of proposals have emerged, using lattice theory, ring theory, number theory and linear algebra. In spite of the incredible potential these developments have for secure cloud computing (and much more), most experts in the field believed that practical implementations (even in customized hardware devices) were many decades away. The WPI Vernam Lab is one of the few teams striving to provide practical implementations of, first, partially homomorphic cryptographic primitives and, perhaps, the world's first practical and secure fully homomorphic encryption system. Project components include number-theoretic optimizations of existing algorithms, a dedicated instruction set for homomorphic operations, and parameter selection and security analysis for specialized lightweight applications." Learn more about the grant
Collaborative Research: Computational Models of Cilia and Flagella in a Brinkman Fluid
National Science Foundation
Motile cilia and flagella are dynamic, elastic, biological structures that exhibit rhythmic motion. Through coordinated beating, cilia in the lung and respiratory tract help to clear the airway of potentially harmful particles and mucus. Impaired cilia motion can result in serious respiratory infection. This impairment can be caused by an altered fluid environment, a characteristic of cystic fibrosis, by defects in the cilia themselves as with primary ciliary dyskinesia, or by other respiratory diseases. Similarly, a sperm is only able to reach and fertilize an egg if its flagellum can propel it forward; impaired motility results in infertility. Cilia and sperm beating is highly dependent on the fluid environment, which contains fibrous, protein networks as well as chemical signals. Understanding this relationship between elastic structure and heterogeneous fluid is vital for development of delivery strategies for inhaled drugs and medicines, new contraceptives, and aiding in infertility due to reduced sperm motility. This project will focus on the development of new computational models and numerical methods that account for these fibrous, protein networks within the fluid to infer in vivo behavior of cilia and sperm. Recent experiments have shown that i) sperm-flagella waveforms are altered when immersed in fluids containing networks of large proteins and also with flagellar calcium concentration, ii) the previously-considered `watery' fluid surrounding airway cilia actually contains a network of large proteins, or, 'brush'. This project will focus on identifying emergent waveforms of sperm flagella and airway cilia when nonplanar bending and internal flagellar biochemistry are taken into account. Cilia and flagella will be modeled as slender, elastic, structures immersed in a fluid governed by the Brinkman equation. A regularized framework and fundamental solutions will be used to derive new numerical methods for various confined geometries. The new numerical methods will be computationally-efficient and developed for use on high-performance computing systems. This research will identify factors that modulate sperm progression towards the egg, explore various waveforms of cilia and sperm, and investigate how a 'brush' in the fluid surrounding cilia might affect transport of particles within that fluid.
The award will also fund RA’ships for two years and provide undergraduate student support. This is part of a 2 PI collaborative proposal, which also include a separate award to Dr. Karin Leiderman at UC Merced.
Learn more about the grant
MRI: Acquisition of a High-Performance Computing System for Research, Education, and Training
National Science Foundation
This award has been used to acquire a high-performance computing (HPC) system for WPI that will enable new levels of research for the investigators on the grant and others, provide new opportunities for education and training at WPI, and be used to raise awareness among young people of the potential of HPC to address problems of societal concern. The system consists of 48 10-core central processing units (CPUs) distributed across 24 nodes. Each node is augmented with two 2496-core high-performance general-purpose graphical processing units (GPUs). Altogether, the system offers 480 CPU cores and 119,808 GPU cores, providing an aggregate peak performance rating of over 51 trillion floating-point operations per second. In both computing power and architectural sophistication, this system greatly exceeds anything previously available at WPI. It will provide a much-needed shared resource for major large-scale computing tasks, an advanced-architecture platform for algorithm development and experimentation, and a highly effective vehicle for education and training in HPC and applications. This HPC system will be used immediately to advance several research projects of the investigators on the grant, including investigations into three-dimensional modeling of sperm motility and interactions, parallel solution methods for coupled multi-block multi-physics systems, computational modeling of human ventricles and plaques, computational validation of effective multi-scale models of thermal behavior in liquid-cooled electronics, and acceleration methods for fixed-point iterations. It will also be made available for research use by other faculty, students, and postdoctoral associates across the university. The investigators' future plans include developing a new graduate course in HPC methods and applications, in which this HPC system will play a central role, and promoting the system's use in existing courses and programs, in particular WPI's new Data Science program and recently developed Bioinformatics and Computational Biology program. WPI's distinguished project-based undergraduate program provides unique opportunities for involving undergraduates in HPC research, and the investigators will jointly develop and advise undergraduate projects that use the HPC system. To further involve undergraduates in HPC applications, they will introduce "real world" industrial projects that use the system to WPI's NSF-funded Research Experience for Undergraduates in Industrial Mathematics and Statistics. To broaden awareness of the role of HPC in science and society, they will develop programs for demonstrations and interactive simulations that will use the system in outreach activities to illustrate how HPC can be used to address problems of societal concern. Also, internship opportunities involving HPC activities will be developed with the WPI-affiliated Massachusetts Academy of Mathematics and Science, a state-wide magnet school for advanced students.
Learn more about the grant.
Optimal Tests for Weak, Sparse, and Complex Signals with Application to Genetic Association Studies
National Science Foundation
Detection of sparse and weak signals is a key for analyzing big data in many fields. Recent statistical research has made celebrated theoretical progress in revealing the detectability boundaries under the Gaussian means model and an idealized linear regression model. Detectability boundary illustrates the border in the two-dimensional phase space of signal sparsity and weakness, below which the signals are asymptotically too weak and sparse to be detectable by any statistical methods. Certain statistics are optimal for these models in the sense that they reach the boundary (i.e., the least requirements) for reliable signal detection. However, there are significant gaps between these theoretical models and practical meaningful models. In this project, the investigators extend statistical theory to handle weak, sparse, correlated, and interactive signals under the framework of generalized linear models. The investigators develop optimal testing procedures to address the realistic data features in genome-wide association studies and next-generation sequence studies. Learn more about the grant.
Preparing Mathematical Sciences Students for Business, Industry, and Government Careers (Pre-BIG)
National Science Foundation
Braddy, L., (PI), Weekes, S., (Co-PI), Dorf, M., (Co-PI), Malek-Madani, R. (Co-PI)
The PreBIG program will provide mathematical sciences faculty with tools and training to help them better prepare students for business, industry, and government (BIG) careers and will provide mathematical sciences students with an opportunity to conduct research on problems related to BIG. To accomplish this, the PIs will produce a set of training videos, conduct summer training workshops for faculty, organize a semester-long course and competition for undergraduate students, organize a summer recognition conference for participating undergraduate students, and secure support from BIG entities. The program includes a strong undergraduate research component since student participation in research has been shown to be effective in improving student success in graduating with a STEM (science, technology, engineering, or mathematics) degree. The undergraduate students will be mentored so that they develop skills that will help them to succeed in a career in STEM, including knowledge of career opportunities, experience in working on problems from BIG organizations, and experience in developing effective writing and oral presentation skills. The MAA's focus on supporting underrepresented groups in the mathematical sciences will be reflected in this program as well. In business, industry, and government, there is a tremendous demand for STEM graduates. Yet, in the mathematical sciences, many students and faculty are unaware of the numerous career opportunities in these sectors, and faculty may not know how to adequately prepare students for STEM careers outside academia. To help remedy this situation, we propose a program to better prepare students in the mathematical sciences to succeed in careers in business, industry, and government (BIG). This program will: a) Increase awareness among mathematical sciences faculty and undergraduate majors of non-academic career options and related internship opportunities. b) Facilitate connections among mathematical sciences faculty and people working in BIG in the same geographic region. c) Offer undergraduate students research opportunities focused on real-world BIG problems. d) Provide training for undergraduate students and faculty in successful approaches to BIG problems along with requisite technical and communications skills. e) Require less and less external funding as time goes on. This project is jointly supported by the Division of Mathematical Sciences and the Office of Multidisciplinary Activities within NSF's Directorate for Mathematical and Physical Sciences.
Learn more about the grant.
New Variational Methods for Quasi-static and Dynamic Material Defect Evolution
National Science Foundation
While defects in materials play a fundamental role in material failure, their analysis remains a major challenge in applied mathematics. This is partly due to the difficulty of formulating precise mathematical models, and partly due to the difficulty of analyzing the free surfaces and singularities involved. The investigator extends recent successes in the analysis of globally minimizing and locally minimizing quasi-static evolutions to both locally stable quasi-static evolutions and dynamic evolutions. One goal is to develop and study new models for cohesive fracture and plasticity with softening, based on local stability rather than global minimality (which is mathematically problematic). The investigator also studies existence and analyzes fundamental properties of dynamic fracture solutions, based on models he formulated previously. The failure of materials rests on the nucleation and evolution of defects such as cracks, plastic regions, and damage. The ability to accurately predict failure depends on the quality of the underlying mathematical models of these defects, as well as on understanding fundamental properties of solutions. Substantial challenges remain in these areas, both in formulating sound models and in the analysis of qualitative behavior of solutions. The investigator seeks to make fundamental progress on these fronts, by developing new models that are both mathematically well-posed and significantly more physically realistic than existing models, and performing the mathematical analysis necessary to assess their accuracy. Learn more about the grant.
Expanding Links with Industry through Collaborative Research and Education in Applied Mathematics
National Science Foundation
The project is a collaborative program of research, education and training based on the Mathematical Problems in Industry (MPI) Workshop and the Graduate Student Mathematical Modeling (GSMM) Camp. The project is part of an ongoing effort organized by the principal investigators at Rensselaer Polytechnic Institute, University of Delaware, Worcester Polytechnic Institute and New Jersey Institute of Technology. These annual meetings, held during successive weeks in June, attract mathematicians (graduate students, postdoctoral fellows and faculty), scientists, and engineers from academic institutions and from industry. The focus of the MPI Workshop is a set of problems brought by contributing participants from industry. These problems span a wide range of areas of applications, often in fluid and solid mechanics but also in mathematical biology, data analysis, and mathematical finance, among others. The scientific objective of the activity generated by the Workshop and its intellectual merit is the study of mathematical problems of significant interest for industrial applications. The GSMM Camp is held during the week prior to the Workshop, and graduate students attending the Camp also attend the Workshop. The main objective of the Camp is graduate student education and training. The two meetings complement each other and form a comprehensive program of interdisciplinary research, education and training that is unique amongst universities in the United States. Learn more about the grant.