1994 — 2001 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Nsf Young Investigator Award @ University of Colorado At Boulder
Cai 9457534 This project is to design and implement fast, scalable, parallel algorthims for the numerical solution of large systems of equations arising from various finite difference/finite elelment/finite volume approximations of linear and nonlinear partial differential equations (PDEs).

1 
1994 — 1996 
Cai, XiaoChuan Mcbryan, Oliver [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Parallel Domain Decomposition Methods and Multilevel Algorithms For Solution of Pdes (Postdoctoral Research Associateship in Computational Science and Engineering) @ University of Colorado At Boulder
McBryan 9406582 The goal of the proposal research is to develop effective parallel sparse linear systems solvers for solving the algebraic equations arising in the finite element discretization of elliptic coupled systems of partial differential equations. These solvers will be based on domain and subspace decomposition methods for use in the solution of linear coupled multicomponent problems, such as the fluidstructure interaction problems. The emphasis will be scalable, transportable algorithms.

1 
2000 — 2004 
Lee, YungCheng (coPI) [⬀] Cai, XiaoChuan Bradley, Elizabeth [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Itr: An Interactive Experimental/Numerical Simulation System With Applications in Mems Design @ University of Colorado At Boulder
This project will build a new generation of numerical simulation systems by creating a feedback path between physical experiments and numerical solvers. There are a number of exciting implications of this dataadaptive simulation idea. Engineering fluid flows are inherently complex. This complexity limits measurement and precision, so engineers are forced to work with fluid flows based on very sparse information. Numerical solvers, on the other hand, can resolve tiny flow structures, but they generally run in an openloop mode and are thus unverified. Coupling the two forms of technology offers powerful advantages to each. Comparisons against live experimental data will allow simulation algorithms to be verified quantitatively, in detail, and inline. Once it is verified in this fashion, one can use the simulation with confidence on related problems. Once can also use the sensor information to correct the solver's data, or even to adjust the solver parameters on the fly. Moreover, once the solver is properly synchronized with the real system, one could use the former to explore the physics of the latter in more detail than sensors would allow  and still trust the results.
A particularly compelling application area for dataadaptive simulation techniques is microelectromechanical systems (MEMS). This emerging technology is driving a revolution in engineering design that is placing new demands on numerical simulation. Accurate modeling of the interaction of tiny, flexible, moving structures with highspeed chaotic fluids is challenging. To resolve the fine details in this kind of simulation, computational fluid dynamics technology requires extremely fine meshes and the solution of very large systems of nonlinear equations. This makes it difficult to build productionquality computeraided design (CAD) tools for MEMS, which in turn forces engineers to fabricate devices without testing them. Functional CAD tools would allow MEMS designers to achieve onepass design, much as VLSI does now.

1 
2000 — 2004 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Parallel Nonlinear Elimination Methods and Software For Partial Differential Equations @ University of Colorado At Boulder
Nonlinear Partial Differential Equations (PDEs) are the basic mathematical description for a wide variety of important application areas. In particular, this project will consider PDEs that arise in fluid dynamics, biology, and radiation diffusion. Because of their complexity, these equations can only be solved numerically by computers, and because of their particular properties (shock waves, sharp fronts, and local singularities) they are difficult to solve even then. This project will design, analyze, and implement software for a class of iterative methods to numerically solve nonlinear PDEs. The software will be provided in two forms  Matlab codes and a package interoperating with the PETSc library  for other researchers to apply the methods.
Technically, the project will study a class of nonlinear elimination algorithms for solving algebraic nonlinear equations with unbalanced nonlinearities. The elimination methods avoid traditional methods' slow convergence when local singularities appear by identifying "misscaled" nonlinear components and replacing them with a function of the remaining more uniformly scaled components. The family of algorithms thus devised will obtain parallelism from domain decomposition, scalability (with respect to problem size) from multilevel methods, and robustness (against local singularities) from incomplete elimination. The methods will be tested on three important classes of applications: transonic compressible flows (CFD), electric wave problems in the heart (computational biology), and Marshak wave problems (radiation transport). The proposed algorithm and software development will have a great impact on the three applications, and will also have substantial influence on other areas of computational science where large nonlinear equations need to be solved.

1 
2001 — 2005 
Cai, XiaoChuan Xi, Yunping (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Itr/Ap: a LiveData Simulation With Application to Bridge Performance @ University of Colorado At Boulder
This project will develop a new generation of numerical simulation systems using advanced parallel computers, mathematical models, and realtime data. Modern sensor technology and the Internet have made it possible to monitor closely the performance of structures such as highway bridges. However, there are often limits on the possible number of embedded sensors, and accurate prediction the overall structural performance therefore requires other technologies, such as computer modeling, at the same time. This project will merge the two technologies, creating a computer modeling system that incorporates the live data in the numerical simulation.
Using measured data as an integrated part of a numerical simulation is a challenging research project. Because data collected by the sensors must be moved continuously into the numerical simulation, the traditional paradigm of reading control parameters at the start of the computation is not possible. Instead, this project will use the ALICE memory snooper from Argonne National Laboratory to allow the constant interruption introduced by the transmission of the measured data. This will in turn allow a parallel computer to exchange information with remote sites without going through slow disk I/O. To fully integrate the sensor data with the computation, the project will develop new numerical schemes based on classical multigrid methods, but using the measured data to build the coarse space. The measured data will also be used as a basis for calibrating and validating the parameters in the mathematical model.
The new simulation system will be used in a high cycle fatigue test of bridge decks, which will be conducted in the Structures Laboratory at the University of Colorado at Boulder. Field tests will also be scheduled on new bridges to be constructed with a variety of installed sensors. These tests are part of ongoing projects sponsored by other agencies. The synergy of these projects will help develop and validate the proposed simulation system.

1 
2001 — 2003 
Shing, P. Benson Farhat, Charbel (coPI) [⬀] Cai, XiaoChuan Willam, Kaspar [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Simulation Platform For the Earthquake Response of Reinforced Concrete Structures @ University of Colorado At Boulder
Motivation: Reinforced Concrete Structures, RCS, are strongly heterogeneous composites, the performance of which depends on the subtle interaction of the brittle concrete and the ductile reinforcement. Under extreme seismic events, the interplay between the two components is critical for the safety and survival of RCS as demonstrated by dramatic failures recent earthquake in Turkey, Taiwan, Japan, and the USA. Objectives: It is proposed to develop a 3dim simulation platform for the earthquake response, SPER, to model progressive failure of RCS subjected to seismic events. The parallel finite element architecture of FEMSFETI, which was developed by the team of Dr. Farhat for the solution of large fluidstructure interaction problems, provides the basic platform for the proposed earthquake simulation environment. SPER will feature a novel nonlinear solver using incomplete elimination, and an innovative interface model to capture localized cracking and shear failure in concrete. For proof of concepts, a RC column will be examined in which quasistatic failure takes place in the form of a cascade of events, such as spalling of the concrete cover, progressive debonding of the reinforcement, yielding and rupturing of the transverse stirrups, and subsequent buckling of the axial reinforcement. As an illustrative example on system analysis, a twospan RC bridge structure will be selected to highlight the 3dim interaction between the bridge superand the pier and to showcase the scalability of SPER. The longterm objective envisions multilevel substructure models of elevated transportation systems exemplified by the Hanshin Express Way, which exhibited dramatic multiple segment failure during the 1995 Kobe earthquake. Methodology: The proposed SPER software will be based on a combination of multilevel/multigrid, domain decomposition and incomplete nonlinear elimination methods. Roughly speaking, in these algorithms, (i) domain decomposition provides the parallelism, (ii) multilevel provides a scalable computing environment with respect to problem size and the number of processors on parallel computers, while (iii) incomplete elimination removes the sensitivity of the nonlinear solver to localized singularities. Relevance: Progressive degradation of RCS is a matter of great importance in earthquake engineering. The current design philosophy has moved to a performancebased approach which requires reliable assessment of the structural behavior beyond the elastic range. SPER is designed to simulate the seismic performance of existing RCS which have been built according to older and often nonconservative earthquake provisions, and to assist the development of rapid and costeffective rehabilitation procedures. Outreach: SPER will be disseminated through a web site which will document the progress and outcome of the modelbased earthquake simulation software of RCS. In the long haul, the intention is to eventually incorporate the 3dim capabilities of SPER into the forthcoming NSF Network for Earthquake Engineering Simulation (NEES), and to develop therein an internetbased capability for earthquake simulations which may be used by colleagues in academia and the earthquake engineering community. Human Resources and Education: The exploratory research will involve three graduate research assistants from the Civil Engineering, Computer Science and Aerospace Engineering, respectively, which will be guided and supervised by the four CoPi's in three departments. They will work in an interdisciplinary environments that encompasses civil infrastructure systems, parallel computing technologies, and numerical solution schemes. The simulation platform SPER will be used as an education tool in a number of undergraduate and graduate courses that are being offered in the Civil Engineering Program at the University of Colorado, such as the Design of Reinforced Concrete Structures, Earthquake Engineering, and Computational Mechanics. SPER will also serve as a demonstration tool for public outreach, especially to high school teachers and K12 students, on earthquake disaster awareness. This effort will take advantage of the new Integrated Teaching and Learning laboratory (ITLL) at the University of Colorado which has received national attention because of its novel approach to learning through discovery.

1 
2002 — 2006 
Cai, XiaoChuan Byrd, Richard [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Itr: Collaborative Research: Optimization of Systems Governed by Partial Differential Equations @ University of Colorado At Boulder
This award addresses an important fundamental problem in computational mathematics: how to optimize complex systems described by partial differential equations (PDEs). The focus is on PDE simulations that can scale into millions of variables, and hundreds or thousands of processors. The size of the problems and the complexity of the techniques for solving these PDEs pose major challenges to modern optimization methods. The project uses a general framework for solving optimization problems in interaction with PDE solvers.

1 
2003 — 2008 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Algorithms: Scalable Solvers For Nonlinear Partial Differential Equations @ University of Colorado At Boulder
The focus of this threeyear research effort is the design, analysis and software implementation of a class of parallel nonlinear iterative methods for the numerical solution of some highly nonlinear partial differential equations (PDEs) arising from important applications in computational fluid dynamics and computational biology. The nonlinear PDEs to be considered in the project are usually not strongly elliptic, and they often contain nonelliptic components causing the solution to be nonsmooth and have local singularities, such as boundary layers or sharp fronts. For smooth nonlinear problems, traditional nonlinear methods, such as Newton's methods, are capable of reducing the global nonlinearities at a nearly quadratic convergence rate but become very slow once the local singularities appear somewhere in the computational domain, even if this happens to only a few components of a largenonlinear system. The proposed algorithm is motivated by the class of nonlinear preconditioning algorithms introduced by Cai and Keyes in 2001 for solving algebraic nonlinear equations that have unbalanced nonlinearities. In nonlinear preconditioning, the global problem is partitioned into subproblems, and subspace nonlinear eliminations are performed on all subproblems. The subproblem are then 'glued' together by a Schwarz type domain decomposition method. Due to the subspace nonlinear elimination, the local singularities are removed, and the global system therefore has more uniform nonlinearity. A family of such algorithms will be studied using a combination of multilevel/multigrid, domain decomposition, nonlinear preconditioning, and nonlinear elimination methods. Roughly speaking, in these algorithms, domain decomposition provides the parallelism, multilevel provides the scalability with respect to the problem size and to the number of processors on parallel computers, and nonlinear elimination removes the sensitivity to the local singularities. Several important application problems will be considered, including the steady state incompressible NavierStokes equations with high Reynolds number and the optimization of a steady state biofluid problem. To study the parallel performance of the algorithms on high performance computers, such as a cluster of workstations and supercomputers, a library will be developed as a plugin package that is fully interoperable with PETSc of Argonne National Laboratory. The proposed algorithm and software development will have a great impact on the application areas, and will also have substantial influence on other areas of computational sciences where large nonlinear equations need to be solved.

1 
2004 — 2009 
Tufo, Henry [⬀] Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Mri: Collaborative Research: Acquisition of An Ibm Bg/L Supercomputer @ University of Colorado At Boulder
This collaborative project among University Corporation for Atmospheric Research, NCAR (0421498 Loft), University of ColoradoBoulder (0420873 Tufo), and University of ColoradoDenver (0420985 Mandel), addressing the technical obstacles to achieving practical petascale computing in geoscience, aerospace engineering, and mathematical applications, aims at acquiring a onecabinet BlueGene/L supercomputer with 2048 compute processors, 64 service processors, and 20 terabytes of storage. Supporting research projects in atmospherics, wildfire modeling, aircraft simulation, and high performance numerical algorithms, the project studies hardware, software, and application issues. A team composed of atmospheric scientists, computational scientists, engineers, applied mathematicians, and experts in mapping applications to highlyparallel systems proposes a tight coupling among computer architects that analyze the performance of computational kernels on new designs (often in isolation) and atmospheric and engineering modelers and designers of scalable solvers proposing new algorithms (often without direct feedback form the computer architects) to extract the highest performance from the next generation of supercomputers. Part of the team has worked with the IBM Watson BlueGene/L development team during the past year and has had access to a BlueGene/L prototype with 512 nodes. Thus, this tightlycoupled group of 12 scientists working together on computational problems, simultaneously considering architectural, algorithmic, and modeling aspects of the problems offers special value to the work.
Broader Impact: The project is expected to accelerate the longterm rate of progress in computational science, which can have a profound impact on the understanding of fundamental scientific questions (such as climate change and wildfires) that are of great importance to society. In addition, providing concrete plans for collaboration and education, the project capitalizes on the large, urban minority population in the CUDenver student body and on the active CUBoulder recruitment plan. Although specific plans are not clear, the PIs all have a record of working with students of diverse backgrounds.

1 
2006 — 2010 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Nonlinear Domain Decomposition Methods and Software For Multicomponent Problems @ University of Colorado At Boulder
The focus of the research is the design and software implementation of a class of parallel nonlinear domain decomposition methods for the numerical solution of highly nonlinear multicomponent partial differential equations arising from important applications in computational fluid dynamics and computational biology. The algorithms and software development will have a great impact on several important application areas, and will also have substantial influence on other areas of computational sciences and engineering where large nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. Current users of PETSc, as well as users of other packages with interfaces to PETSc, will be able to access to the algorithms and software.
These multicomponent problems are usually not elliptic, and these nonelliptic components often cause the solution to have local singularities, such as boundary layers or sharp fronts. The new algorithms are motivated by the class of nonlinear elimination algorithms and nonlinear preconditioning algorithms for solving algebraic nonlinear equations with unbalanced nonlinearities. In the new algorithms the misscaled nonlinear components are first identified, then using the implicit function theorem of calculus these components are replaced by a function of the remaining more uniformly scaled components. The original system is hence reduced implicitly to contain only the smooth components. Several important classes of applications are considered including problems in traditional CFD and problems in the emerging field of computational biology. To study the parallel performance of the algorithms on high performance computers a library will be developed as a plugin package that is fully interoperable with PETSc of Argonne National Laboratory.

1 
2007 — 2012 
Cai, XiaoChuan Xi, Yunping (coPI) [⬀] Zane, Regan (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Noss: An Integrated Power Aware SensorSimulation Network System For LongTerm Performance Assessment of Concrete Infrastructures @ University of Colorado At Boulder
The performance of civil infrastructures plays an important role in maintaining the quality of life of the U.S. citizens as well as in the nation's defense systems. The project attacks several critical issues involved in the new generation of smart civil infrastructure system with builtin sensor network. Performance assessment of civil infrastructure is usually a longterm task (i.e. several years) the power supply for the sensor network becomes a problem, especially for the embedded sensors in structures. The first focus is to provide a reliable energy harvesting method for the sensors and a power management plan for the sensor network. For longterm performance assessment, processing the large amount of data obtained from the sensor network is a difficult and important task, the second focus is how to configure the sensor network and assemble the measured data in a more meaningful way in order to use the available information to predict future structural performance. To this end, this project is to develop an integrated Power Aware SensorSimulation Network system for longterm performance assessment of concrete infrastructures. The system consists of three subsystems including an energy harvesting and power management system for the sensors, a combined sensorbased numerical simulation system, and a massively parallel computing system for the numerical calculations. Builtin sensors have found their applications not only in civil, and aerospace structures, but also in mechanical engineering and medical devices such as artificial hearts. Therefore, the research will have a great impact in many areas of science and technologies.

1 
2008 — 2010 
Overton, Michael [⬀] Keyes, David Szyld, Daniel Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Scalable Parallel Algorithms For Partial Differential Equations
Scalable Parallel Algorithms for Partial Differential Equations Abstract Scalable algorithms for partial differential equations play a key role in Computational Science and Engineering. As the number of processors in today's supercomputers grows rapidly, scalability becomes one of the most important issues in algorithm design. Tremendous progress has been made in this research area in the past few years. The conference brings together experts in this field from all over the world to discuss the latest development of scalable algorithms, and to introduce junior researchers to this exciting research field. The conference also supports the participation of graduate students and provides them with a unique opportunity for exposing their research to a large audience and for learning from experts. The primary focus is on domain decomposition METHODS which COMPRISE by far the most common scalable paradigm for largescale simulation on massively parallel distributed, hierarchical memory computers. In domain decomposition, a large problem is reduced to a collection of smaller problems, each of which is easier to solve computationally than the undecomposed problem, and most or all of which can be solved independently and concurrently. Typically, it is necessary to iterate over the collection of smaller problems, and much of the theoretical interest in domain decomposition algorithms lies in ensuring that the number of iterations required is very small. Indeed, the best domain decomposition methods share with their cousins, multigrid methods, the property that the total computational work is linearly proportional to the size of the input data, or that the number of iterations required is at most logarithmic in the number of degrees of freedom of individual subdomains. Algorithms whose work requirements are linear in the size of the input data in this context are said to be optimal. Optimal domain decomposition algorithms are now known for many, but certainly not all, important classes of problems that arise in science and engineering. Much of the current research interest in domain decomposition algorithms lies in extending the classes of problems for which optimal algorithms are known. Domain decomposition algorithms can be tailored to the properties of the physical system as reflected in the mathematical operators, the number of processors available, and even to specific architectural parameters, such as cache size and the ratio of memory bandwidth to floating point processing rate.

0.954 
2009 — 2013 
Li, Congming (coPI) [⬀] Cai, XiaoChuan Ge, Shemin (coPI) [⬀] Williams, Mark (coPI) [⬀] Williams, Mark (coPI) [⬀] 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Cmg Research: Multiscale Nonlinear Domain Decomposition Method For Modeling the Impact of Climate Change On Groundwater Resources @ University of Colorado At Boulder
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 1115).
CMG Research: Multiscale Nonlinear Domain Decomposition Method for Modeling the Impact of Climate Change on Groundwater Resources
S. Ge, Department of Geological Sciences, University of Colorado X. Cai, Department of Computer Science, University of Colorado C. Li, Department of Applied Mathematics, University of Colorado M. Williams, Department of Geography, University of Colorado
Continuing climate change poses uncertainties on future water resources. The water cycle encompasses fundamental processes that link various elements of climate and water resources. Holding approximately 30% of Earth's fresh water, groundwater?s enormous storing capacity can be an effective buffer in regulating more drastic hydrologic events on the surface, therefore, plays an important but often overlooked role in longterm sustainability of water resources. High altitude mountainous regions are vital source areas for water. Hydrologic processes in highaltitude regions are particularly sensitive to climate change because of the presence of snow, glaciers, and permafrost. Yet, basic questions remain regarding how groundwater is replenished at its source by mountain recharge, the size of groundwater reservoirs, as well as how permafrost influences groundwater. Modeling the groundwater flow processes involving mountain recharge and permafrost faces mathematical challenges due to nonlinearity of the governing equations and multiscale nature of the spatial and temporal domains.
The objective of the research is to develop a more accurate mathematical model and a new robust computational algorithm and high performance software to study the impact of climate change on groundwater resources in mountain watersheds, with a focus on quantifying mountain recharge and permafrost hydrology. The research plan is to first develop a mathematical model that will be capable of handling coupled fluid flow and heat transport in complex geologic systems in multiscale spatial and temporal domains. Second, field hydrogeologic study at two sites will be conducted to gather data for testing the mathematical model. The final stage is to conduct numerical simulations to assess the response of groundwater storage and flow in mountain watersheds to future climate change scenarios.
First, this study will contribute to our scientific knowledge on water cycle processes at multi spatial and temporal scales. In particular, this study will make a unique contribution to strengthening the subsurface element of the water cycle, increasing knowledge on mountain recharge, and integrating little known permafrost hydrology into a water resource study. Second, highly parallel and robust numerical algorithms and software will be developed for the coupled multiphysics system describing the water cycle processes. Third, the proposed mathematical model development will be a substantial contribution to hydrogeologic sciences. Dealing with multiscale fluid flow problems in heterogeneous geologic media has been a long standing challenging. Development of a robust computational algorithm allows efficiently modeling of hydrogeologic systems at such a comprehensive and integrated level that would be difficult to achieve by either mathematicians or geoscientists alone.
Water resource sustainability and climate change are pressing issues of global and local concern. This study will benefit long term planning of water resources and increase general public?s knowledge on the linkage between climate and water resources, by dissimilating results through local media and public lectures. The new computational algorithm implemented by a robust and versatile software will be transferable to other areas of application and available to other researchers. The crossdiscipline nature of this project will afford students in mathematics and geosciences a unique opportunity to interact with each other in intellectual and physical settings that differ from those they are used to. This will be achieved by requiring students to take classes outside their home departments, math students to participate in field work, geoscience students to be trained in computational mathematics. Finally a joint mathgeosciences seminar will be established to involve all project personnel. By encouraging broad participation, this seminar will foster more and sustained future collaborations between mathematics and geosciences.

1 
2009 — 2014 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Nonlinear Preconditioning Techniques For Coupled MultiPhysics Problems On Massively Parallel Computers @ University of Colorado At Boulder
Mature technologies are available for solving many types of single physics problems, but for coupled multiphysics problems, robust and scalable techniques are badly needed, especially for large scale parallel computers. The focus of the proposal is on some new domain decomposition based nonlinear preconditioning techniques for the numerical solution of some highly nonlinear, coupled systems of partial differential equations (PDEs) arising from multiphysics applications. These PDEs often represent multiple interacting fields (for example, fluid and solid), each is modeled by a certain type of equations. Current approaches usually involve a careful splitting of the fields and the use of fieldbyfield iterations to obtain a solution of the coupled problem. Such approaches have many advantages such as ease of implementation since only single field solvers are needed, but also exhibit disadvantages. For example, certain nonlinear interactions between the fields may not be fully captured, and for unsteady problems, stable time integration schemes are difficult to design. In addition, when implemented on large scale parallel computers, the sequential nature of the fieldbyfield iterations substantially reduces the parallel efficiency. To overcome the disadvantages, fully coupled approaches are investigated in order to obtain full physics simulations. The success of such a fully coupled approach depends almost entirely on a nonlinear algebraic system solver that is robust and scalable. Unfortunately, traditional nonlinear iterative methods do not work well, for example, Newtonlike methods often converge very slowly because of the existence of local nonsmooth components in the solution and the lack of good initial guess. The new algorithms are motivated by the nonlinear preconditioning methods recently introduced by the PI and his coworkers for solving algebraic nonlinear equations that have unbalanced nonlinearities. The scalability is obtained by incorporating the multigrid methods into the algorithms. Several important applications will be studied including the simulation of blood flows in compliant arteries using a coupled NavierStokes and elasticity equations.

1 
2012 — 2016 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Af: Small: Fully Implicit Methods For Partial Differential Equations and Software For Hybrid Architecture @ University of Colorado At Boulder
The first goal of the project is to develop new domain decomposition algorithms and software for the numerical solution of some highly nonlinear, coupled systems of partial differential equations arising from multiphysics applications. For simple problems, implicit methods are relatively easy to develop from a given explicit or semiimplicit method, but for some multiphysics problems it is quite difficult to develop a fully implicit method that allows a high performance implementation. Most of the existing techniques are nonsmooth and therefore difficult to solve with Newton type solver. In the project, some discretization techniques will be developed that involve high order nonsmooth discretization for capturing the domain information, and low order locally smooth discretization for building the Jacobian system of the Newton iterations. The low order discretization serves as a nonlinear preconditioner that speeds up the convergence, but doesn't change the accuracy of the solution. The second goal of the project is to develop an efficient implementation of the proposed linear and nonlinear preconditioning approaches on high performance computers with a large number of processors.
Relatively mature technologies including algorithms and software are available for solving many types of single physics problems, but for coupled multiphysics problems, robust and scalable techniques are badly needed, especially for large scale parallel computers with accelerators. The proposed algorithms and software will have a great impact on several important application areas, such as the simulation of global atmospheric flows and the biofluids, and will also have substantial influence on other areas of computational sciences where large linear and nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. The research is a rich area in opportunities for both graduate and undergraduate students interested in high performance computing and general Computational Science and Engineering.

1 
2017 — 2020 
Cai, XiaoChuan 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Parallel Nonlinear Preconditioning Algorithms and Applications in Biomechanics @ University of Colorado At Boulder
Computer simulation of fluidstructure interaction problems has many applications in science and engineering, such as the vibration analysis of aircraft, automobiles, and suspension bridges. More recently, the technique has been extended to diagnosing and treatment planning of certain medical problems such as congenital and acquired cardiovascular diseases, and to the design and optimization of medical devices. Solving the fluidstructure interaction problems on supercomputers with a large number of processor cores is challenging because the mathematical model consists of a coupled complicated nonlinear system, and most existing algorithms and software for the solution of problems of this kind do not scale well beyond a few hundred processor cores. The principal investigator will develop highly scalable algorithms and software that are suitable for large scale supercomputers and applicable for different models of blood flows and material parameters for the arterial wall. Mature technologies are available for solving many types of linear problems, but for coupled, highly nonlinear multiphysics problems, robust and scalable techniques are badly needed, especially for implementation on large scale parallel computers. The technical focus of this project is a class of nonlinearly preconditioned Newton methods that combines a nonlinear elimination technique with multilevel domain decomposition for parallelization. Through this research the principal investigator will solve nonlinear difficult problems modeling a wide range of physical models with different levels of nonlinearities. Algorithms that provide a high degree of parallelism will be designed so that large scale parallel computers can be used efficiently. The target application is a family of fluidstructure interaction problems in biomechanics. For the fluid model, both Newtonian and nonNewtonian models will be studied. For solid models, linear elasticity will be considered for small deformations, geometric nonlinear elasticity will be considered for large deformations, and hyperelasticity will be considered for materially nonlinear problems. Two and manylevel versions of the algorithms will be investigated to obtain high scalability on parallel computers with a large number of processor cores. This research will have a great impact in areas of computational science and engineering where nonlinear difficult problems need to be solved. The research is rich in opportunities for both graduate and undergraduate students interested in applications in biomechanics, parallel computing, and general computational science and engineering.

1 