Ricardo H. Nochetto - US grants

This research will consider the numerical analysis of strongly nonlinear problems including free boundary problems, constrained problems such as Stokes flow, and the Hamilton-Jacobi equations. Linearization techniques based on nonlinear Chernoff formulae will be investigated, and nonlinear algorithms for both parabolic and eliptic problems will be analyzed with particular emphasis on accuracy in nonenergy spaces and splitting algorithms. The impact of local mesh refinements in reducing computational labor for a given accuracy will be considered. Work analyzing the pointwise accuracy of mixed finite element methods will be continued. This work will combine and interrelate theory and numerical methods. It should increase understanding of both the numerical techniques and the motivating physical situations.

With this award the principal investigator will continue his studies into the numerical analysis of phase transitions and free boundary problems that arise in fluid dynamics and elasticity. In particular, he will analyze, design and implement adaptive finite element methods for the solution of the Phase Field Model and the Cahn-Hilliard equation. Special emphasis will be placed upon stability and error analyses in nonenergy norms, such as the maximum norm, and upon the efficiency and reliability of the resulting algorithms. Many phenomena in nature involve what are called free boundary problems, wherein a large part of the solution process is devoted to determining the location of a moving boundary between two or more different media. A good example is a weather forecast or the weather map in your newspaper. Here meteorologists try to predict the speed and location of fronts; in this context fronts are the words used for moving boundaries between warm and cold air, for example. With this the principal investigator will design numerical methods for finding the solution of some free boundary problems that come from the theory of elasticity. **

This project will support research in the design, implementation, and analysis of adaptive finite element methods for phase transitions and free boundary problems arising in material sciences. Numerical methods for phase transitions will be studied; the use of highly graded meshes is extremely important for phase transitions and free boundary problems which typically exhibit sharp interfaces as well as very thin transition regions. The solution, or some of its derivatives, may exhibit discontinuities or just vary very rapidly within such layers, leading to global numerical difficulties if the singularities are not properly resolved. Related sharp interface models exhibiting competition between surface tension and destabilizing mechanisms will also be studied, such as stress driven instabilities in solid crystals and solidification processes. The design of linearization techniques for strongly nonlinear partial differential equations will be studied and combined with mesh refinements. The interaction of convective fluid with free boundaries, for example that corresponding to solid coexisting with its melt and interfaces of multiphase fluids, will be analyzed numerically. Finally, mixed methods for constrained problems will be studied, with special emphasis on the Stokes flow and related free boundary problems as well as on free boundary problems for plates over obstacles. This project is concerned with the design and implementation of adaptive mesh refinements for nonlinear partial differential equations arising in thermodynamics, fluid dynamics, and elasticity. The use of highly graded meshes is important for many of these problems, which typically exhibit sharp interfaces as well as very thin transition regions. The increasing interest in such problems stems not only from their mathematical features but also their applications to phase transitions in material sciences, flame propagation, combustion theory, and crystal growth.

The adequate numerical treatment of nonlinear phenomena governed
by partial differential
equations with several
disparate scales is a formidable computational challenge. Modern
algorithms should be able to resolve fine scales for certain physical
quantities without overresolving others, thereby optimizing the
computational effort and making realistic 3d simulations feasible.
Phase transitions in materials science, epitaxial growth,
viscous incompressible fluids, and chemotaxis
are typical yet quite distinct examples.
The goal of this project is to design, test, and
analyze reliable and efficient adaptive finite element methods
for such problems, with
space-time error control and based on
refinement/coarsening mesh modification.
This project builds
upon our novel theory for a posteriori
error estimation of evolution problems; in fact it extends and
enhances the prior NSF Grant
DMS-9623394. It blends quite delicate analytical and computational
issues, and applies them to free boundary problems, constrained
problems, and advection-diffusion equations.

This project has strong connections with areas of strategic interest
such as Materials Science, Micro Electronics and High-Performance
Computing, in that it addresses central issues to all of them
such as the development of reliable and
efficient algorithms.
This is a collaborative project involving a number of scientists
in the US and abroad, as well as several students.
A substantial effort is devoted to education
and human resource development.

0129243
Nochetto
This award supports Ricardo H. Nochetto and students from the University of Maryland-College Park in a collaboration with Gerhard Dziuk of the Department of Applied Mathematics at the University of Freiburg, Germany. The award is a one-year renewal of a project begun in January 2000. The project focuses on error control for nonlinear partial differential equations via a posteriori error estimation. The techniques will have applications for physical problems involving diffusion, advection, phase changes, and interface dynamics. Such problems arise frequently in industrial applications, and raise fundamental and challenging questions that need to be addressed in order for the field to advance. The collaboration combines complementary expertise and facilities on both sides. In particular the innovative software toolbox ALBERT, developed by the German group, is central to the proposed work, which will gain added value from the powerful theoretical capabilities of the U.S. group. The work plan provides for extensive participation by graduate students in the international travel and research.

INTERPHASE is a series of meetings, devoted mainly to numerical and computational aspects of free boundary problems, which have been held successfully once a year in Europe for the past 8 years. The apparent misspelling of INTERPHASE is supposed to emphasize the chief purpose of this meeting: facilitate the communication and exchange of ideas and methods between communities (`phases') of scientists and engineers that otherwise may not interact and thus profit from each other. In fact, interfaces arise in a variety of applications from materials science and fluid dynamics, to image processing and finance, which may not have any common roots and goals but however are described by similar partial differential equations and thereby present analogous difficulties, challenges, and methodology.

The 9th conference of the series will be held in College Park, MD, in September 2001 with the intention of encouraging more participation of US researchers and stimulating the exchange of ideas between American and European peers in the area of problems where there is an interface between regions of interest. We plan to have about 30 senior speakers, 12 junior speakers, and a poster session.

The adequate numerical treatment of nonlinear phenomena
governed by partial differential equations with several disparate
scales is a formidable computational challenge. Modern algorithms
should be able to resolve fine scales for certain physical
quantities without overresolving others, thereby optimizing the
computational effort and making realistic 3d simulations
feasible. Epitaxial and crystal growth in materials science, and
viscous incompressible fluids are typical yet quite distinct
examples addressed in this proposal. The goal of this project is
to design, test, and analyze reliable and efficient adaptive
finite element methods for such problems, with space-time error
control and based on refinement/coarsening mesh modification.
This project blends quite delicate analytical and computational
issues, and applies them to free boundary problems, constrained
problems, geometric PDE and the Navier-Stokes equations of
incompressible fluids.
Scientific computing has joined theory and experiment to
form together the three central aspects of scientific inquiry.
The current strengths in computational mathematics draw on the
widespread acceptance of computational modeling as a complement
to, and even a replacement for, physical tests in a broad number
of fields. In this vein, the investigator develops reliable and
efficient computational tools that may be useful in several areas
of strategic importance such as nanotechnology, materials
science, and high-performance computing. This project is a
collaborative endeavor, involving a number of scientists in the
US and abroad, as well as several students and postdocs. A
substantial effort is devoted to education and human resource
development.

0126272
Nochetto

This US-Argentina award will fund a collaborative project between Dr. Ricardo H. Nochetto, University of Maryland, College Park (UMCP), in collaboration with Drs. Hugo A. Aimar and Eleonar O. Harboure, Instituto de Matematica Aplicada del Litoral (IMAL), Universidad Nacional del Litoral in Santa Fe, Argentina, and Dr. Carlos Cabrelli of FCEyN, Universidad de Buenos Aires in Argentina. Their research is a mix of classical and functional analysis as applied to the practical, numerical solution of partial differential equations (PDE). The problem is one of approximating solutions by means of mathematical objects that can be computed.

This project will build upon the strengths and complementary interests of each group, to collaborate on the topics of harmonic and numerical analysis. UMCP has expertise on adaptivity for linear and nonlinear PDE as well as basic wavelet theory and applications. The IMAL has expertise on wavelet applications to PDE and Besov spaces. They will facilitate an ongoing collaboration by bringing both sides together for six visits over two years. In addition, they will include advanced graduate students and postdocs. This pairing will promote the cross-fertilization of ideas and techniques between the two research areas, institutions and countries.

The adequate numerical treatment of nonlinear phenomena governed by partial differential equations (PDE) with several disparate space-time scales is a formidable mathematical and computational challenge. Modern algorithms should be able to resolve fine scales for certain physical quantities without overresolving others, thereby optimizing the computational effort and making realistic three-dimensional simulations feasible. This proposal deals with fundamental mathematical questions for the design, testing, and analysis of adaptive finite element methods (AFEM) as well as their application to a variety of multiscale problems for which AFEM are among the most powerful computational techniques.

This project considers mathematical models in materials science (such as epitaxial and crystal growth), in biophysics (such as biomembranes), in fluid and solid mechanics (such as the Navier-Stokes equations), in image processing and in finance. They are typical, yet quite distinct, examples of multiscale phenomena which exhibit singularities, fast transients, and topological changes.

This proposal builds upon, and in fact extends and enhances, the prior NSF Grant DMS-0204670. It is organized in a number of small and seemingly independent projects, which are however related through the interplay of nonlinearity, error estimation, numerical analysis and computation, the unifying themes of the proposal. It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. Resources are requested to support them partially. A substantial effort is devoted to education and human resource development.

Nochetto
DMS-0807811

The adequate numerical treatment of continuum phenomena
exhibiting disparate space-time scales is a formidable
mathematical and computational challenge. Modern algorithms
should be able to resolve fine scales for certain physical
quantities without overresolving others, thereby optimizing the
computational effort and making realistic 3d simulations
feasible. Mathematical models in biophysics (such as
biomembranes in both fluid and gel state), in materials science
(such as crystal surface morphologies and bilayer actuators), and
in shape optimization (including both fluid and solid mechanics)
are typical yet quite distinct examples addressed in this
project. Their multiscale structure manifests in rapid
transients and slow motion regimes in time, as well as point and
line singularities (interfaces) and thin layers in space. Large
domain deformations, perhaps leading to topology changes, is
another intriguing feature. The goal of this project is to
design, test, and analyze reliable and efficient adaptive finite
element methods for these multiscale problems, most governed by
geometric partial differential equations, with space-time error
control based on a posteriori error estimation. This project
builds upon, and in fact extends and enhances, the work of the
prior NSF Grant DMS-0505454. It is organized in a number of
small but interrelated topics permeated by the roles of geometry
and adaptivity throughout, from issues in numerical analysis and
computation to applications.

This project blends quite delicate analytical,
computational, and modeling issues in several areas of research
of strategic importance such as nanotechnology, materials and
manufacturing, biotechnology, and high performance computing.
Designing smart electro-mechanical devices as well as
understanding key functions of lipid membranes in living
organisms, both at the nano and microscales, require powerful and
flexible computational algorithms capable of dealing with large
shape variations. This project develops such tools. It is a
collaborative and interdisciplinary project involving a number of
scientists in the US (four of them engineers and physicists) and
abroad, as well as several students and postdocs. Because a
substantial effort is devoted to education and human resource
development, most resources are used to partially support
students and postdocs.

0754983
Shapiro

Electrowetting is a technique for manipulating fluids on the micro-scale. By applying voltages at actuating electrodes, it is possible to (effectively) modify surface tension properties, and to move, split, merge, and mix liquid packets. Applications of electrowetting include re-programmable lab-on-a-chip systems, auto-focus cell phone lenses, and colored oil pixels for laptops and video-speed smart paper.

The PIs will develop experimentally validated models that will predict electrowetting dynamics first in two, then in three, spatial dimensions, which will enable next-generation system analysis, design, and control. The models will include the essential bulk-flow physics: surface tension, low-Reynolds fluid dynamics, electrostatics or electrodynamics, as well as critical loss-phenomena such as contact angle saturation and hysteresis. Moving liquid/gas or liquid/liquid interfaces, that can undergo split and merge topology changes, will be tracked by a combination of an implicit finite element (FEM) method, which will allow computation of interface curvature and the resulting surface tension forces naturally, easily, and accurately, and by the level-set method applied only locally at split/merge events, will naturally yield topology changes. This will combine the strengths of FEM (it handles curvature extremely accurately) and the level-set approach (it naturally captures topology changes). FEM will also be used to solve the low-Reynold's Navier Stokes equations, the electrostatic (or electrodynamic) part of Maxwell's equations, and to handle boundary conditions at the moving solid/liquid/gas triple line in a numerically sound manner. Triple line motion/pinning models will be evaluated and compared against electrowetting experiments - this will improve the initial hysteresis model and will incorporate a combined hydrodynamic and averaged molecular-kinetic description from the literature.

Intellectual Merit
Currently, there are no modeling tools to understand and quantify the dynamic behavior of electrowetting systems. To build such models, the PIs will: 1) include the essential physical phenomena, 2) correctly state the bulk partial-differential-equations (especially the interplay between electrodynamic effects and the resulting fluid forces), 3) use the variational method to recast these equations and then create numerically viable FEM algorithms to solve them, 4) track moving interfaces, that can undergo topological changes, by a combination of the FEM and level-set methods, in a numerically sound manner, 4) include loss-phenomena such as saturation and hysteresis from first-principles and the literature (when possible) or from experimental data (when not), and 5) validate against electrowetting experiments, by isolating and confirming each new part. The merit is in achieving and combining these components.

Broader Impact
The PIs collaborate with two leading electrowetting groups (at a university and a company), and are about to begin a collaboration with a third (a company). All three groups have expressed a strong need for such a physical-first-principles, experimentally informed, dynamic electrowetting modeling tool. If successful, the results will be used by the electrowetting community to understand, analyze, design, and control next-generation electrowetting systems. The methods developed for tracking 2-phase micro-flow topology changes will be of use in many other micro-fluidic applications.

Nochetto
DMS-1109325

Capturing the essential behavior of nonlinear phenomena at the micro- and nanoscales with the simplest and crudest models is of fundamental importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of ensuing processes, and the analysis of both models and algorithms. Mathematical models in biophysics (such as biomembranes in both fluid and gel state), in materials science (such as crystal surface morphologies and bilayer actuators), and in shape optimization (including electro-wetting on dielectric) are typical yet quite distinct examples that the investigator studies in the project. The governing partial differential equations are geometric and exhibit disparate space-time scales: point and line singularities (interfaces), thin layers, and large domain deformations, perhaps leading to topology changes. The goal of the project is to model and control such multiscale phenomena, and to design, test, and analyze reliable and efficient adaptive finite element methods for them with space-time error control based on a posteriori error estimation.

Understanding the mechanisms of nonlinear phenomena at micro- and nanoscales is essential in many areas of science and engineering. The investigator develops mathematical models and reliable computational methods for studying a wide range of such problems. This project deals with applications of Federal strategic interest such as nano and microtechnology (such as the design and control of micro electro-mechanical system (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. A substantial effort is devoted to education and human resource development.

Capturing the essential behavior of nonlinear phenomena with the simplest possible models is of paramount importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of devices, and the analysis of both models and algorithms. These crucial aspects of modern research are blended together in this research project, which deals with modeling, formulation, and numerical analysis of physical and biological phenomena at a scale where surface tension competes with bulk effects and could in principle be manipulated (or controlled) to produce scientifically interesting and practically useful dynamical behavior. Applications of the work include nano and microtechnology (such as the design and control of micro electro-mechanical systems (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). Results of the work will enhance modeling and prediction capabilities and help educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of research.

This project investigates models, such as biomembranes, ferrofluids, liquid crystals, and bilayer actuators, that are governed by nonlinear geometric partial differential equations defined on deformable domains that are unknown beforehand. Numerical approximation is carried out via adaptive finite element methods, with a posteriori error estimation and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The project will advance understanding of adaptive approximation methods and the role of geometry in key questions concerning:
1. Convergence and complexity of adaptive finite element methods (FEM) for elliptic PDE; study of fractional diffusion, hybridizable discontinuous Galerkin methods, hp-FEM and isogeometric methods, and the Laplace-Beltrami operator on parametric surfaces.
2. Design of high order arbitrary Lagrangian-Eulerian methods for parabolic PDE on deformable domains and surfaces.
3. Control of problems involving surface tension and magnetic effects, with or without free boundaries, relevant for device design in technology and biomedicine.
4. Computational modeling and analysis of ferrofluids and liquid crystals; these are technologically useful and mathematically intriguing complex fluids which can be actuated by magnetic and electric fields, and thus manipulated and controlled for specific purposes.
5. Novel FEM for geometric PDE: handling of large deformations with isometry constraints, typical of bilayer actuators, and dealing with fully nonlinear PDE.

The International Conference on the Foundations of Computational Mathematics 2017 is the ninth of a successful series and will be held in Barcelona, Spain from July 10 to July 19, 2017. This award funds the attendance at the meeting of at least twenty junior U.S. participants, including graduate students, postdoctoral fellows, and early career tenure track faculty. The conference features seventeen plenary speakers, eight of whom are from the US, and 21 workshops where the junior participants participants will be able to present a talk or a poster.

The impact of constantly increasing computational power has changed the relationship between mathematics and computation. New applications such as encryption, large scale data management, and signal/image processing, inevitably led to the study of the computational tools themselves, creating and reinvigorating a rich spectrum of mathematical disciplines. Mathematicians now increasingly recognize that, besides its value as a numerical and experimental tool, computation is a significant theoretical tool in its own right. This conference aims to further the understanding of the connections between mathematics and computation, including the interfaces between pure and applied mathematics, numerical analysis and computer science. This is reflected in the diverse and vibrant research areas covered by the 21 workshops which are organized by world experts, including many from the US. This award is instrumental in enabling the participation of many US young researchers at this renowned and exciting conference.

Conference website: http://www.ub.edu/focm2017/

Fabrication and manipulation of new and smart materials, particularly in the strategic areas of nanotechnology and biotechnology, require understanding of nonlinear phenomena governed by geometric partial differential equations (PDEs). At small scales, say micro and nano scales, surface tension and bending effects dominate bulk effects, thereby making the actuation and control of small devices a reality. This leads to scientifically interesting and technologically useful configurations and dynamic behavior. Examples abound in biomedical sciences (drug delivery vesicles, cell encapsulation devices, and sensors) and engineering (photovoltaic devices, optics, energy storage, micromotors, microgrippers, microvalves, and adaptive deformable mirrors). However, microfabrication is time-consuming, expensive, and often erratic, which makes the development of predictive computational tools of paramount importance in engineering and science. This project deals with modeling, analysis, and computation of geometric problems of interest in materials science, biophysics, plasma physics, and robotics. It enhances modeling and prediction capabilities and helps educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of contemporary research.

Capturing the essential behavior of nonlinear phenomena with the simplest and crudest models is fundamental in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of devices, and the analysis of both models and algorithms. These crucial aspects of modern research are incorporated into the following four intertwined projects: geometric PDEs with constraints (bilayer actuators and prestrained films, shape optimization for plasma confinement, and fully nonlinear PDEs); actuation of complex fluids (liquid crystals actuated by electric fields and temperature, and ferrofluids actuated by magnetic fields); nonlocal models (efficient solvers for linear and nonlinear fractional diffusion and stochastic control); a posteriori error analysis and adaptivity (high-order methods, fractional PDEs, and free boundary problems). Numerical treatment of nonlinear geometric PDEs is a formidable scientific challenge due to the dynamic deformation of geometries, the presence of strong nonlinearities, and the development of self-penetrating structures and topological changes. Efficient algorithms should optimize and balance the computational effort and thus capture small scales without over-resolving others, thereby leading to accurate interface description. This project develops structure-preserving finite element methods (FEMs) with a posteriori error control (adaptive FEMs) and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The roles of geometry, nonlinearity, nonlocality, and adaptive approximation permeate the research, from basic questions in numerical analysis of nonlinear PDEs to applications in strategic areas of national interest. Graduate and postdoctoral students participate in the research of the project.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.