Carlos J. Garcia-cervera - US grants

In a series of research works we have introduced and established the
positivity principle for schemes for solving hyperbolic systems of conservation
laws. The rationale of the positivity principle is stability, which is a very
important requirement for numerical schemes. The positivity principle is the
1st stability principle for schemes for solving multi-dimensional hyperbolic
systems. In this proposal we have shown that the central scheme studied by Kurganov and Tadmor is positive. By mixing upwind scheme and Lax-Wendroff scheme, we have made a positive scheme which costs only 30% of the original positive scheme. We have developed a scheme called Convex Essentially Non-Oscillatory (ENO) scheme. The Convex ENO scheme is a high order accurate central scheme. We have developed a new multigrid method to solve hyperbolic systems of conservation laws. By doing multigrid, the cost of calculations is reduced significantly. In the proposal we also develop several schemes for solving elliptic problems with multi-fluids separated by the interfaces. Such problems arise from many real world applications. For example, incompressible multi-fluids Navier-Stokes equation. A new uniform 2nd order accurate scheme on non-body-fitting grids is developed for that. We have proposed a uniform 2nd order accurate level-set method using finite element method for solving elliptic problems with mixing boundary conditions. Such problems emerge from in simulating epitaxial thin film growth using the island dynamics model. We have used some of those methods to do Direct Numerical Simulation on multi-phase turbulent flows. We have developed a geometric multigrid method for such elliptic problems based on the Ghost Fluid Method, and plan to do more with the other methods. The PI and his collaborators are pursuing further development of positive schemes. In a series of research works they have introduced and established the positivity principle for schemes for solving hyperbolic systems of conservation laws. The rationale of the positivity principle is stability. 1) They prove that the central scheme developed by Kurganov and Tadmor is positive scheme. 2) They continue to develop a new positive scheme, which is a mixture of upwind and Lax-Wendroff schemes. Hence two-stage Runge-Kutta is not required and for two-dimensions the computation cost could be cut by as much as 70%.
3) They continue to work on a new scheme called weighted component-wise positive
scheme. It is a mixture of Weighted ENO schemes and 2nd order component-wise version of Convex ENO scheme or high-resolution central scheme. They use a convex combination of all candidates to do reconstruction, but use a new measurement called accurateness instead of smoothness to assign proper weights. The convex combination achieves almost optimal (one order lower than the optimal) order accuracy. This scheme can be also extended to solve Hamilton-Jacobi equations in multi-dimensions. 4) They are going to introduce a multigrid method for solving multi-dimensional hyperbolic systems of conservation laws. The novelty is to calculate the fluxes on coarse grid, then interpolates the differences of the fluxes or the fluxes to the finest grid. Such multigrid method is not only faster than a base scheme in each iteration, but also allows larger time step than that of the base scheme. Hence the multigrid method requires much less CPU time to advance solutions to the same stopping time compared to the base scheme. In other words, for the same CPU time, the multigrid method advances solutions much further in time. This is particularly useful for computing stationary solutions. In the recent years, the PI and his collaborators have been pursuing further development of Ghost Fluid Method (GFM) for multi-phase fluids. 1) They propose a geometric multigrid method to solving linear systems arising from irregular boundary problems involving multiple interfaces in 2D and 3D. In this method, they adopt a matrix-free approach i.e. they do not form the fine grid matrix explicitly and they never form nor store the coarse grid matrices. The main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function. 2) They propose a 2nd order accurate level-set method using finite element method for solving elliptic equations with Robin interface conditions. They first study a weak formulation of it, and then prove that
there exists a unique weak solution. At last, a finite element method on non-body-fitting uniform or arbitrary triangulations is used to solve such weak formulation. The novelty of this work is the incorporation of finite element methods and non-body-fitting triangulations. 3) They develop a new 2nd order accurate numerical method on non-body-fitting grids for solving the elliptic equations with interfaces. Instead of smooth, the boundary and the subdomains'
boundaries and hence the interfaces, are only required to be Lipschitz continuous as submanifold. A weak formulation is developed and the numerical method is derived by discretizing the weak formulation by piece-wise linear functions. The method is 2nd order accurate in maximum norm if the interface is smooth or its discontinuities are proper handled, and convergent in maximum norm otherwise. 4) They use the boundary condition capturing method
to do Direct Numerical Simulations on multi-phase turbulent
flows. This is the first successful DNS of such problems.
Because turbulence happens through a large range of scales, and hence very efficient methods are needed to capture all meaningful scales.

The proposal focus on the real world applications. For example, hyperbolic
systems of conservation laws, incompressible Navier-Stokes equations with interfaces, epitaxial thin film growth using the island dynamics model, Direct Numerical Simulation on multi-phase turbulent flows. The proposed numerical methods possess high order accuracy and high resolutions, hence they are very efficient. Two multigrid methods are proposed to couple with those methods to further speeded up numerical simulations. The proposal should have broad impact, since the methods created can be easily adopted to many other application areas in the environmental, geophysical, biological, material science, and engineering sciences.

The investigator takes up three projects. In the first
project, the emphasis is the analysis and the design of efficient
numerical methods for the simulation of recent models proposed for
the dynamics of the magnetization coupled to the spin accumulation
in multilayers when a current perpendicular to the plane of the
layers is applied. The dynamical equation for the spin is a
diffusion equation with discontinuous coefficients, and the
magnetization dynamics is described by the Landau-Lifshitz
equation from ferromagnetism. In the second project, the
investigator studies a new model for liquid crystals based on
Inhomogeneous Density Functional Theory. The model can describe
the isotropic, nematic, smectic A, and crystalline phases.
Finally, in the third project, the investigator studies the
formation of aggregates in polymer mixtures.

Ferromagnetic materials are widely used in the magnetic
recording industry, which continues to be the dominant technology
for the storage of digital data. The project addresses some of
the technological difficulties found in the design of magnetic
devices such as magnetic memories (MRAMs). Liquid crystals have a
wide range of applications. Their optical properties are highly
sensitive to electric and magnetic fields, which has made liquid
crystals ideal candidates for optical-switching devices, such as
video displays and optical memories. They are very sensitive to
temperature changes as well, which makes them useful in
applications involving sensors. More generally, polymers appear
in most aspects of our life, from the amniotic fluid inside
biological cells, to plastics and shampoos used on a daily basis.
Understanding the dynamics of biological membranes and transport
through them is of fundamental importance in the development of
efficient methods for the delivery of medicines. Understanding
the process of aggregation in polymer mixtures can have a
significant impact in the design of new products. This study of
polymers and liquid crystals addresses some of these technological
and biological issues.

The performance of nanometer-sized devices can be greatly affected by the formation and propagation of defects, such as cracks. Given that this process involves breaking the chemical bonds in the solid lattice, a quantum-mechanical description is necessary. However, a fully atomistic description of the problem is not viable. The PI proposes to develop analysis-based numerical methodologies that couple an atomistic description near the location of the defect with a coarse-grained description far from it. Quantum-mechanical effects are studied in the context of Density-Functional Theory. This approach can effectively remove the limitation in the size of the system considered, and could gear the development of nanometer-sized devices in new directions.

This proposal also includes the development of special courses in multiscale modeling, and an undergraduate research component in a vertical environment that involves undergraduate students, graduate students, and post-doctoral fellows. In addition, the proposal includes outreach to local high schools. By exposing students to mathematics at such an early stage, the investigator hopes to awaken in them the desire to pursue a career in mathematics.

Joo
DMS-0908538

The investigator and her colleague study mathematically the
interaction of smectic liquid crystals with external electric or
magnetic fields. Typical liquid crystals are nematic and smectic
liquid crystals, consisting of rod-like molecules that tend to
align themselves lengthwise along a common axis, called the
director. The local orientation of a nematic liquid crystal is
described by the director field. However, in smectic phases the
molecules of the liquid crystal also tend to lie in layers. To
describe the local orientation, a smectic order parameter must be
introduced in addition to the director field. The free energy of
smectic liquid crystals has a resemblance to the Ginzburg-Landau
energy for superconductivity, and it is a nonlinear, nonconvex
second order energy. While nematic phases are rather well
understood, the mathematical theory of smectic phases is
comparatively underdeveloped. This study consists of two main
projects. In the first project, the investigators advance the
mathematical understanding of smectic liquid crystals in the area
of instabilities driven by external magnetic fields, by means of
mathematical analysis and numerical simulations. Partial
differential equation methods for phase transition, calculus of
variations, Gamma convergence theory, and bifurcation theory are
employed. In the second project, the investigators study
ferroelectric liquid crystals such as smectic C* and bent-core
phases, which are named after the bent or banana-shaped
molecules. The response time of bent-core liquid crystals is
only in the range of microseconds and thus this material offers
huge potential for many applications. The investigator studies
an electrodynamic model of the molecular reorientation of
ferroelectric liquid crystals.

Liquid crystals are used in many different applications such
as optical switching devices and temperature sensors. Nematic
liquid crystal materials have been widely studied because of
their applications to flat panel displays. However, it is known
that ferroelectric liquid crystals such as smectic C phases with
chiral molecules exhibit much faster switching times. Thus
understanding the features of smectic liquid crystals such as
defect structures and switching dynamics is a very important
issue in practical applications.

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."
Proposal #: 09-60316
PI(s): Brown, Frank, L.; Fredrickson, Glenn, H.; Garcia-Cervera, Carlos; Gilbert, John, R.; Van de Walle, Christian, G.
Institution: University of California-Santa Barbara
Title: MRI-R2: Acquisition of a High Performance Central Computing Facility at UCSB
Project Proposed:
This project, acquiring of a computational cluster to replace a six-year-old system, allows access to fast mid-sized parallel computation to dozens of researchers and serves as the institution's primary resource for parallel computation. The system is structured for a variety of uses. Standard MPI computation is carried out with a tightly coupled cluster of quad core processors, while 'fat nodes' with 256 GB of RAM as well as local high speed disk storage service jobs that require large shared memory. Researchers actively developing codes can take advantage of the unique performance characteristics of GPU (graphics processing) nodes. The system will have several NVidia Tesla nodes.
The users of the system are drawn from all five departments of the College of Engineering (Chemical Engineering, Computer Science, Electrical & Computer Engineering, Materials, and Mechanical Engineering), seven departments of the Division of Mathematical, Life and Physical Sciences (Chemistry & Biochemistry, Earth Science, Ecology Evolution & Marine Biology, Mathematics, Molecular Cellular & Developmental Biology, Physics, and Psychology), as well as the departments of Economics, Geography, and Media Arts & Technology from the Division of Humanities and Social Sciences. In addition, the system supports research in eight campus centers: Allosphere Research Facility, the California NanoSystems Institute, the Center for Polymers and Organic Solids, the Institute for Crustal Studies, the Kavli Institute for Theoretical Physics, the Materials Research Laboratory, the National Center for Ecological Analysis & Synthesis, and the Neuroscience Research Institute.
The new system will be housed in the same location as the previous one where the same successful administrative and maintenance procedures used for the past six years will be applied. The proposed cluster will be accessible via the UC Grid, a web portal interface that makes high performance computing resources easy to use from desktop machines (PCs or Macs). The acquired system will come with a three-year warranty. Prior experience has shown that only a small number of nodes are expected to malfunction during the useful lifetime (> 3 years) of the cluster.
Broader Impacts:
The research enabled by the campus-wide facility, interdisciplinary and collaborative in nature, is available to the broad research community. The large majority of users, roughly 75%, consists of postdocs, graduate students, and undergraduates (5%), allowing this award to accomplish NSF's longstanding goal of integrating research and education. Outreach to K-12 takes place via a new initiative, The School for Scientific Thought (SST), an extension to the Let's Explore Physical Sciences (LEAPS) Program. Under the SST Program UCSB science and engineering graduate students design and teach a course for an audience of high school students on Saturdays. In addition, many of the faculty associated with this proposal participate in the UC Leadership Excellence through the Advanced Degrees program that has increased the number of underrepresented students in science and engineering at UCSB.

TECHNICAL SUMMARY:

One of the most important scientific challenges is how to efficiently harvest, convert, store and utilize solar energy. In recent years, there has been a growing interest of developing organic materials for solar cell applications. Organic solar cells offer a low-cost, large-area, flexible, light-weight, clean, and quiet alternative energy source for both indoor and outdoor applications. However, their power conversion efficiencies and operational lifetimes must be improved to enable large-scale commercialization and implementation and deep societal impact. Thus, there is an urgent need to understand fundamental processes in these devices. Currently, the synthesis and optimization of new materials is time consuming and labor intensive, and relies on trial and error approaches with poor success rates. There is therefore a great need to rationally anticipate materials performance from their chemical composition and bulk morphology to accelerate technology development and depart from empirical optimization. The goal of this interdisciplinary program is to mesh complementary expertise in chemistry, materials, physics, and mathematics, to make breakthroughs in the science and technology of organic solar cells. The team will address: 1) the development of new methods to simulate phase separation in BHJ solar cells; 2) the development of first-principles methods to predict carrier mobilities in organic semiconductors; 3) the nanoscale characterization of donor-acceptor interpenetrating networks; 4) the understanding of charge generation and transport process; 5) the synthesis of new conjugated polymers guided by the theoretical predictions; 6) evaluation of materials performance. The simulation methods will be validated extensively on well-studied materials and will then be capitalized for the design of more efficient new materials. As well-orchestrated theoretical and experimental efforts, the project strives to achieve transformative breakthroughs for the development of high-efficiency and low-cost organic solar cells.

NON-TECHNICAL SUMMARY:

The world demand for energy is expected to double by 2050. As of now, there is no viable technology to address this challenge without emission of carbon dioxide to the environment. In view of this, increasing the power conversion efficiency and operational lifetime of plastic solar cells is provides the opportunity to create a clean and potentially economically viable energy source with a wide range of applications. The goal of the proposed research is to assemble the team of scientists with complementary expertise in chemistry, materials, physics, and mathematics, to establish theoretical guidelines for a rational development of materials and solar cell device structures. Successful completion of the program is expected to relieve the current need to optimize device performance via trial and error procedures and will pave the way for an acceleration of technical innovation. This project integrates interdisciplinary research and education by involving the participation of undergraduate and graduate students and postdoctoral researchers, with special emphasis in the recruitment of underrepresented students. New courses on organic semiconductors and their applications in energy conversion will be offered at UCSB. The courses will be taught in an interdisciplinary environment, with which the investigators hope to address the urgent need to train researchers in the area of renewable energies. Furthermore, Workshops and demonstrations on solar energy for K-12 students, teachers, and parents in local schools will be developed to create awareness about renewable and sustainable energy sources. In addition, by exposing students to these ideas at such an early stage, the investigators hope to awaken in them the desire to pursue a career in sciences.

The quantum physics of many interacting electrons lies at the foundation of chemistry and condensed matter physics. A direct treatment of the many-electron problem is impossible due to its shear complexity: dealing with N interacting electrons requires solving partial differential equations in 3N dimensions. Equilibrium and non-equilibrium Density Functional Theories (DFT) are rigorous and formally exact theories which map the interacting N-electron problem into a non-interacting N-electron problem. The non-interacting electrons move in an effective potential that has a universal functional dependence on the total electron density. As a result, the problem is reduced to a problem in dimension 3, amenable for computation. In this proposal the PIs propose to study a number of dynamical problems in many-body quantum mechanics within an interdisciplinary environment of mathematicians and physicists. In particular, the PIs propose to develop further the mathematical foundations of density-functional theory, for equilibrium as well as the time-dependent case. The mathematical structure of the theory and its solutions will be further investigated and the insight from this analysis will be used to develop efficient numerical simulations. Particular emphasis will be given to the treatment of the spin-orbit interaction, within the full relativistic formulations and in non-relativistic formulations that include relativistic corrections. The PIs also plan to establish the foundations of the Dissipative Time-Dependent Density Functional Theory, and to apply the theory to the problem of charge and spin transport in materials.

The present technological progress is in great part based on design and discovery of new materials. Nowadays, the design of advanced materials involves laboratory work and computer simulations. Enhancing the accuracy and efficiency of computer simulations will reduce the costs, broaden the array of interesting and potentially useful materials, and speed up the process of testing and characterization. This is the target of the proposed research. The plan is to combine rigorous mathematical analysis, the insights from physics, chemistry and computer simulations in order to push the boundaries of theoretical simulations of advanced materials such as nano-structured materials, topological insulators and molecular electronic devices. The proposed research could have significant technological impact in applications such as nano-science and other areas of interest such as solar cell devices and energy conversion and storage. The PIs propose to integrate research and education by involving undergraduate and graduate students, and post-doctoral associates, in an interdisciplinary environment. Special attention will be paid to the recruitment of women and students from other underrepresented groups through the utilization of a diverse number of programs at the participating institutions.

The main objective of the present proposal is to develop a new model for multiscale analysis of solids, the quasi-atomistic model, which plays a transitional role between the atomistic and continuum models. In addition, the PIs propose to develop an energy-based procedure to couple fully atomistic descriptions and continuum descriptions in a seamless way, which allows for systematic coarse-graining with controllable error. A second objective is the analysis of the proposed methodology, including consistency, stability and convergence. Inconsistency, known as ghost force in this context, can be systematically removed even in the case of interfaces with corners. Another component of the project is to combine the principles of the multigrid method and multi-resolution analysis for the efficient implementation. This will be done by constructing multilevel grids by multi-resolution analysis and solving nonlinear minimization problems with multigrid methods.

Molecular mechanics is a common approach to modeling the behavior of matter, where atoms are treated as the essential degrees of freedom and a potential function is used to describe the interactive effects between atoms. Equilibrium structures can be computed by minimizing the potential energy with respect to the position of the atoms. Other information about the system, such as the vibrational spectrum, thermodynamic properties, equations of state, and reaction rates, can also be computed. Extensive applications of molecular mechanics modeling can be found in materials science, chemistry and biology. The proposed work will mainly focus on solids, with generalizations to soft matter and biological systems, and conduct mathematical, numerical and applicative studies of the proposed methodology.

This project, acquiring a computer cluster (mini-supercomputer) to replace the old one, aims to serve the computational needs facilitating scientific research and education in multiple areas. The machine includes 120 node Infiniband interconnected cluster for efficient message passing interface (MPI) parallel processing, four shared memory "fat nodes" with 1 Terabyte (TB) of memory/node, four graphic processing unit (GPU) nodes built around NVIDIA Tesla P100 1 Gigabyte (GB) GPUs, and four Intel Knight's Landing nodes. This blended system will serve the computational needs of the vast majority of campus researchers. This system will also service users needing large-scale resources by allowing development, prototyping, and benchmarking calculations locally, prior to production runs at supercomputer centers.

The research enabled spans multiple departments and supports research in many campus centers including AlloSphere Research Facility, the California Nanosystems Institute, the Center of Polymers and Organic Solids, the Earth Research Institute, the Kavli Institute of Theoretical Physics, the Marine Science Institute, the Materials Research Laboratory, and the National Center for Ecological Analysis and Synthesis. This interdisciplinary and mainly collaborative research will be facilitated by the presence of an available local facility, without the administrative hurdles and delays associated with application to supercomputing centers.

Broader Impacts:
The cluster is expected to play a prominent role in educating the next generation of scientists, engineers, and mathematicians. The NSF Integrated Graduate Education and Research Traineeship (IGERT) program in Network Science and Big Data will also employ the cluster that will additionally be utilized by undergraduates, high school, and community college students and K-12 teachers via existing sponsored programs. It will also contribute to attract others students and researchers. Furthermore the facility will service graduate participants in the University of California Leadership Excellence through Advanced Degrees (UCLEADS) and the Bridges to Doctorate programs. These programs help increase the number of under-represented students involved.