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A communication-avoiding 3D sparse triangular solver

We present a novel distributed memory algorithm to improve the strong scalability of the solution of a sparse triangular system. This operation appears in the solve phase of direct methods for solving general sparse linear systems, Ax = b. Our 3D sparse triangular solver employs several techniques, including a 3D MPI process grid, elimination tree parallelism, and data replication, all of which reduce the per-process communication when combined. We present analytical models to understand the communication cost of our algorithm and show that our 3D sparse triangular solver can reduce the per-process communication volume asymptotically by a factor of O(n1/4) and O(n1/6) for problems arising from the finite element discretizations of 2D "planar" and 3D "non-planar" PDEs, respectively. We implement our algorithm for use in SuperLU_DIST3D, using a hybrid MPI+OpenMP programming model. Our 3D triangular solve algorithm, when run on 12k cores of Cray XC30, outperforms the current state-of-the-art 2D algorithm by 7.2x for planar and 2.7x for the non-planar sparse matrices, respectively.

Sao, Piyush↗

Scalable Adaptive Multilevel Solvers for Multiphysics Problems

In this project, we carried out many studies on adaptive and parallel multilevel methods for numerical modeling for various applications, including Magnetohydrodynamics (MHD) and complex fluids. We have made significant efforts and advances in adaptive multilevel methods of the multiphysics problems: multigrid methods, adaptive finite element methods, and applications.

97 MATHEMATICS AND COMPUTING↗

Mechanisms of interfacial reactivity in near surface and extreme environments

The local water structure surrounding ions in aqueous solutions greatly affects their chemical properties such as reaction rates, ion association, and proton and electron transport. These properties result in the behavior of ions in natural aqueous environments. For example ore transport is facilitated by chloride ion pair formation and the reaction of ions in an interface is strongly dependent on the dehydration of the ion hydration shell. We are developing the use of high-­resolution XAFS observations and 1st principles based MD-­XAFS analysis (spectra simulated using 1st principle methods with no adjustable parameters, AIMD) to interpret the solution properties of strongly interacting aqueous solutes under arbitrary pressure and temperature conditions. In the 1st principle MD-­XAFS method density functional theory (DFT) based MD simulations(Car and Parrinello, 1985) are used to generate a large ensemble of structural snap shots of the hydration region. These are then used to generate scattering intensities. I emphasize three points about this novel approach to analyzing XAFS data. 1st: As illustrated in Figure 1, the level of agreement between the calculated and observed intensities is considerably higher than has been obtained by any XAFS analysis to date (note 2nd shell region, R> 2 Å). 2nd: This result was obtained from a parameter free simulation with no fitting of the interaction potentials to any data. This supports the use of these methods for more difficult environments and more complex solutes (polyions). 3rd: New information about the shell structure (Figure 1) is now available because of this more detailed agreement. We note also that both multiple scattering and second shell features are well represented in the analysis. As far as we know this is the 1st analysis of second shell structure and multiple scattering. Excellent agreement has been obtained for most of the third row metal ions: Ca 2+ , Zn 2+ , Cu 2+ , Ni 2+ , Co 2+ , Mn 2+ , Fe 3+ , Cr 3+ . Calculations on these systems are demanding because of their open electronic shells, and high ionic charge. Principal Investigator: Professor John Weare (University of California, San Diego) The prediction of the interactions of geochemical fluids with minerals, nanoparticles, and colloids under extreme near surface conditions of temperature (T) and pressure (P) is a grand challenge research need in geosciences (U.S. DOE 2007, Basic Research Needs for Geosciences: Facilitating the 21st Energy Systems.). To evaluate the impact of these processes on energy production and management strategies it is necessary to have a high level of understanding of the interaction between complex natural fluids and mineral formations. This program emphasizes 1st principle parameter free simulations of complex chemical processes in solutions, in the mineral phase, and in the interfaces between these phases The development of new computational tools (with emphasis on oxide materials and reaction dynamics) tailored to treat wide range of conditions and time scales experienced in such geochemical applications is has been developed. Because of the sensitivity of the interaction in these systems to electronic structure and local bonding environments, and of the need to describe bond breaking/formation, our simulations are based on interactions calculated at the electronic structure level (ab-initio molecular dynamics, AIMD). The progress in the computational aspects of program may be summarized in terms of the following themes (objectives); Development of efficient parameter free dynamical simulation technology based on 1st principles force and energy calculations especially adapted for geochemical applications (e.g., mineral, interfaces and aqueous solutions) (continuing program); Calculation of the dynamics of water structure of in the surface-water interface of transition metal oxides and oxihydroxides; and Development of improved (beyond DFT+GGA) electronic structure calculations for minerals and the interface region that more accurately calculate electron correlation, spin density, and localization. The focus of the program is also on the iron oxide and oxihydroxide minerals and Fe 2+ (aq)/Fe 3+ (cr) oxidation in the mineral solution interface region. These methods included the development of model Hamilitonian methods that can be solved to near convergence for single site models (DMFT) and many-body perturbation methods (MP2, GW); Development of time decomposition methods to extend time scales of molecular dynamics (MD) simulations and support the use of high complexity electronic structure calculations (MP2, CCSD(T)) of forces for use in dynamical simulations where very high chemical accuracy is required (microsolvated reactions in absorbed surface layers); and The development of a new linear scaling finite element solver for eigenvalue problem that supports solution of quantum problems with unusual potential and boundary values. Application progress of the above new simulation technology to problems of geochemical interests includes; The prediction of metal oxide surface structure and the reduction/oxidation of Fe 3+ (cr)/Fe 2+ (aq) in metal oxide (hematite, goethite)/solution interfaces. Result: water interacts strongly with the 001 Hematite surface; interaction of water with the 100 goethite is weak; The study of ion solvation and the composition of ion hydration shells under extreme conditions (focus on Fe 3+/2+ , Al 3+ and Mg 2+ and their hydroxide speciation). Result: Ion association in water solutions can be calculated from 1st principle methods. Efficient sampling of the free energy requires more development; The continued development of new high resolution analysis of XAFS scattering of disordered systems (particularly Al, Mg) and of XANES calculations for aqueous ions. Result: EXAFS spectra can be calculated to high accuracy with DFT level dynamic simulations; The exploration of electron localization and electron transport in metal oxides (highly correlated materials). Result: Proper description of electron localization requires levels of calculation beyond DFT; and Localization of electrons in DFT type Hamiltonians was studied. Result: For very Dirac high exchange new solutions (New unphysical bifrucations) to the eigenvalue problem are found. The program was highly collaborative involving faculty and students in mathematics, physics and computer science departments as well as coworkers at the Pacific Northwest National Laboratories (PNNL). The students in this program had the opportunity to develop skills in the development of methods, the implementation of method on high performance parallel computers and the application of these methods to problem in geochemical science. Much of the software that was developed was incorporated in the NWchem software package maintained by PNNL.

58 GEOSCIENCES↗

ALE3D: An Arbitrary Lagrangian-Eulerian Multi-Physics Code

ALE3D is a multi-physics numerical simulation software tool utilizing arbitrary-Lagrangian- Eulerian (ALE) techniques. The code is written to address both two-dimensional (2D plane and axisymmetric) and three-dimensional (3D) physics and engineering problems using a hybrid finite element and finite volume formulation to model fluid and elastic-plastic response of materials on an unstructured grid. As shown in Figure 1, ALE3D is a single code that integrates many physical phenomena.

42 ENGINEERING↗

Building an Efficient Model for Afterburn Energy Release

Many explosives will release additional energy after detonation as the detonation products mix with the ambient environment. This additional energy release, referred to as afterburn, is due to combustion of undetonated fuel with ambient oxygen. While the detonation energy release occurs on a time scale of microseconds, the afterburn energy release occurs on a time scale of milliseconds with a potentially varying energy release rate depending upon the local temperature and pressure. This afterburn energy release is not accounted for in typical equations of state, such as the Jones-Wilkins-Lee (JWL) model, used for modeling the detonation of explosives. Here we construct a straightforward and efficient approach, based on experiments and theory, to account for this additional energy release in a way that is tractable for large finite element fluid-structure problems. Barometric calorimeter experiments have been executed in both nitrogen and air environments to investigate the characteristics of afterburn for C-4 and other materials. These tests, which provide pressure time histories, along with theoretical and analytical solutions provide an engineering basis for modeling afterburn with numerical hydrocodes. It is toward this end that we have constructed a modified JWL equation of state to account for afterburn effects on the response of structures to blast. The modified equation of state includes a two phase afterburn energy release to represent variations in the energy release rate and an afterburn energy cutoff to account for partial reaction of the undetonated fuel.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization

This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on prevalent applications in computational mechanics and design optimization. We leverage Matérn-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models as well as the shape and topology of optimized designs. Here we describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications, emphasizing the potential to influence these domains. The flexibility and efficiency of SPDE-based GRF generation empowers us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate design features and topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model and quantify geometric uncertainties on reconstructed submanifolds, such as the interpolated surfaces of cerebral aneurysms provided by postprocessing CT scans. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.

97 MATHEMATICS AND COMPUTING↗

Hermite analogs of the lowest order Raviart–Thomas mixed method for convection–diffusion equations

The Raviart–Thomas mixed finite-element method of the lowest order (Raviart and Thomas in mixed finite-element methods for second-order elliptic problems, Lecture Notes in mathematics, Springer, New York, 1977) commonly known as the RT{sub 0} method, is a well-established and popular numerical tool to solve diffusion-like problems providing flux continuity across inter-element boundaries. Douglas and Roberts extended the method to the case of more general second-order boundary-value problems including the convection–diffusion equations (cf. this journal Douglas in Comput Appl Math 1:91–103; 1982). The main drawback of these methods, however, is the poor representation of the primal variable by piecewise constant functions. The Hermite analog of the RT{sub 0} method for treating pure diffusion phenomena proposed in Ruas (J Comput Appl Math, 246:234–242; 2013) proved to be a valid alternative to attain higher order approximation of the primal variable while keeping intact the matrix structure and the quality of the discrete flux variable of the original RT{sub 0} method. Non-trivial extensions of this method are studied here that can be viewed as Hermite analogs of the two Douglas and Roberts’ versions of the RT{sub 0} method, to solve convection–diffusion equations. A detailed convergence study is carried out for one of the Hermite methods, and numerical results illustrate the performance of both of them, as compared to each other and to the corresponding mixed methods.

97 MATHEMATICS AND COMPUTING↗

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗

Regression Based Approach for Robust Finite Element Analysis on Arbitrary Grids. LDRD Final Report

This report summarizes the work performed under a one-year LDRD project aiming to enable accurate and robust numerical simulation of partial differential equations for meshes that are of poor quality. Traditional finite element methods use the mesh to both discretize the geometric domain and to define the finite element shape functions. The latter creates a dependence between the quality of the mesh and the properties of the finite element basis that may adversely affect the accuracy of the discretized problem. In this project, we propose a new approach for defining finite element shape functions that breaks this dependence and separates mesh quality from the discretization quality. At the core of the approach is a meshless definition of the shape functions, which limits the purpose of the mesh to representing the geometric domain and integrating the basis functions without having any role in their approximation quality. The resulting non-conforming space can be utilized within a standard discontinuous Galerkin framework providing a rigorous foundation for solving partial differential equations on low-quality meshes. We present a collection of numerical experiments demonstrating our approach in a wide range of settings: strongly coercive elliptic problems, linear elasticity in the compressible regime, and the stationary Stokes problem. We demonstrate convergence for all problems and stability for element pairs for problems which usually require inf-sup compatibility for conforming methods, also referring to a minor modification possible through the symmetric interior penalty Galerkin framework for stabilizing element pairs that would otherwise be traditionally unstable. Mesh robustness is particularly critical for elasticity, and we provide an example that our approach provides a greater than 5x improvement in accuracy and allows for taking an 8x larger stable timestep for a highly deformed mesh, compared to the continuous Galerkin finite element method. The report concludes with a brief summary of ongoing projects and collaborations that utilize or extend the products of this work.

97 MATHEMATICS AND COMPUTING↗

A numerical approach for simulating fluid structure interaction of flexible thin shells undergoing arbitrarily large deformations in complex domains

We present a new numerical methodology for simulating fluid–structure interaction (FSI) problems involving thin flexible bodies in an incompressible fluid. The FSI algorithm uses the Dirichlet–Neumann partitioning technique. The curvilinear immersed boundary method (CURVIB) is coupled with a rotation-free finite element (FE) model for thin shells enabling the efficient simulation of FSI problems with arbitrarily large deformation. Turbulent flow problems are handled using large-eddy simulation with the dynamic Smagorinsky model in conjunction with a wall model to reconstruct boundary conditions near immersed boundaries. The CURVIB and FE solvers are coupled together on the flexible solid–fluid interfaces where the structural nodal positions, displacements, velocities and loads are calculated and exchanged between the two solvers. Loose and strong coupling FSI schemes are employed enhanced by the Aitken acceleration technique to ensure robust coupling and fast convergence especially for low mass ratio problems. The coupled CURVIB-FE-FSI method is validated by applying it to simulate two FSI problems involving thin flexible structures: 1) vortex-induced vibrations of a cantilever mounted in the wake of a square cylinder at different mass ratios and at low Reynolds number; and 2) the more challenging high Reynolds number problem involving the oscillation of an inverted elastic flag. For both cases the computed results are in excellent agreement with previous numerical simulations and/or experiential measurements. Grid convergence tests/studies are carried out for both the cantilever and inverted flag problems, which show that the CURVIB-FE-FSI method provides their convergence. Finally, the capability of the new methodology in simulations of complex cardiovascular flows is demonstrated by applying it to simulate the FSI of a tri-leaflet, prosthetic heart valve in an anatomic aorta and under physiologic pulsatile conditions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Developing highly scalable fluid solvers for enabling multiphysics simulation.

We performed an investigation into explicit algorithms for the simulation of incompressible flows using methods with a finite, but small amount of compressibility added. Such methods include the artificial compressibility method and the lattice-Boltzmann method. The impetus for investigating such techniques stems from the increasing use of parallel computation at all levels (processors, clusters, and graphics processing units). Explicit algorithms have the potential to leverage these resources. In our investigation, a new form of artificial compressibility was derived. This method, referred to as the Entropically Damped Artificial Compressibility (EDAC) method, demonstrated superior results to traditional artificial compressibility methods by damping the numerical acoustic waves associated with these methods. Performance nearing that of the lattice- Boltzmann technique was observed, without the requirement of recasting the problem in terms of particle distribution functions; continuum variables may be used. Several example problems were investigated using a finite-di erence and finite-element discretizations of the EDAC equations. Example problems included lid-driven cavity flow, a convecting Taylor-Green vortex, a doubly periodic shear layer, freely decaying turbulence, and flow over a square cylinder. Additionally, a scalability study was performed using in excess of one million processing cores. Explicit methods were found to have desirable scaling properties; however, some robustness and general applicability issues remained.

Clausen, Jonathan R↗

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Mathematics↗

Automated Model Updating Effect on the Linear and Nonlinear Dynamic Responses of Historical Masonry Structures

Finite element models supply the simulation of the problems to obtain the structural behaviors under different loading conditions. The updating processes based on changing the uncertain parameters such as material properties, boundary conditions, meshing size etc. are made to create actual finite element models in order to represent the real conditions of the problems. This procedure can be made using two methods, namely manual model updating and global/local automatic model updating. The global/local automatic model updating procedures are very popular owing to the applicability to all kind of engineering structures, minimizing the differences nearly to zero, availability of damage localization and structural health monitoring. In this paper a historical masonry armory building selected as an application and its finite element analyses, experimental measurements and automated model updating processes are carried out. The finite element model of the structures was developed with ANSYS software and first three natural frequencies are attained. The experimental dynamic characteristics such as natural frequencies, damping ratios and mode shapes are obtained using ambient vibration method. The differences between experimental and numerical frequencies are minimized up to the range of 0–1%, by using local updating procedure. The linear and nonlinear seismic analyses of the building are conducted on initial and updated models. At the end of the analyses, maximum displacements, principal stresses and strains are given with detail. In addition, crack distributions diagrams are attained to validate the stresses accumulation points. It can be seen that there is an increasing trend in the displacement, stress and strain values after finite element model updating both linear and nonlinear analysis. In addition, linear and nonlinear analysis results are compared with each other to evaluate the nonlinear analysis effect. It is concluded that the displacement and strain values are increased after nonlinear analysis. But, there is not any agreement in stress values.

42 ENGINEERING↗

COMPARISONS OF THE FINITE-ELEMENT-WITH-DISCONTIGUOUS-SUPPORT METHOD TO CONTINUOUS-ENERGY MONTE CARLO FOR PIN-CELL PROBLEMS

The standard multigroup (MG) method for energy discretization of the transport equation can be sensitive to approximations in the weighting spectrum chosen for cross-section averaging. As a result, MG often inaccurately treats important phenomena such as self-shielding variations across a material. From a finite-element viewpoint, MG uses a single fixed basis function (the pre-selected spectrum) within each group, with no mechanism to adapt to local solution behavior. In this work, we introduce the Finite-Element-with-Discontiguous-Support (FEDS) method, whose only approximation with respect to energy is that the angular flux is a linear combination of unknowns multiplied by basis functions. A basis function is non-zero only in the discontiguous set of energy intervals associated with its energy element. Discontiguous energy elements are generalizations of bands and are determined by minimizing a norm of the difference between snapshot spectra and their averages over the energy elements. We begin by presenting the theory of the FEDS method. We then compare to continuous-energy Monte Carlo for one-dimensional slab and two-dimensional pin-cell problem. We find FEDS to be accurate and efficient at producing quantities of interest such as reaction rates and eigenvalues. Results show that FEDS converges at a rate that is approximately first-order in the number of energy elements and that FEDS is less sensitive to weighting spectrum than standard MG.

Cross sections↗

Hybrid Optimization Software Suite

The hybrid optimization software suite (HOSS) is a general purpose fully 2D/3D parallel combined finite discrete element (FDEM) code that can be used to simulate problems involving fracture and fragmentation processes, large deformation and large rotations, discrete particle systems, solid-fluid interaction in hydro fracture problems, etc. HOSS uses a hybrid approach that combines finite-element and discrete-element methods with a novel computational fluid dynamics solver. The finite-element method is often used to analyze a material or object and how it responds to stress. The discrete-element method analyzes stresses and displacements in a volume containing a large number of particles, such as grains of sand. Fluid dynamics analyzes the fluid flow inside, around, or through solid domains. With these processes combined, HOSS represents a paradigm shift when it comes to generating accurate simulations of material deformations and failure. HOSS can perform the following: Resolves problems that consist of millions of deforming fracturing, and interacting solid particles; Requires no coupling—it naturally integrates with all regimes of fluid flow; Resolves all regimes of fluid flow, such as Stokes, low/high Reynolds, subsonic/transonic/supersonic/hypersonic, compressible/incompressible, viscid/inviscid, Newtonian/Non-Newtonian and turbulent/laminar flow; Combines all flow regimes in the same solver; Numerically stable for all flow regimes; Naturally combines different flow regimes in the same problem; Possesses material library for fluid behavior; Naturally matches fluid solver to solid solver; Includes non-inertial Eulerian formulation for fluids; Uses highly efficient parallel computing based on the Virtual Parallel Machine, thus enabling easy porting between different parallel architectures, including possible future architectures; Designed specifically for multi-physics problems in research and industry in virtual experimentation format, thereby complementing both theoretical-computational and experimental tools; Can be extended to address millions of deforming solid particles immersed in fluid flowing at all regimes; Can be extended to address biological applications, such as modeling bone fracture or red blood cells in blood plasma

Knight, Earl E.↗

A finite element formulation for deformation twinning induced strain localization in polycrystal magnesium alloys

Deformation twinning induces shear strain localization in hexagonal close-packed crystals and is critical for the material’s ductility and failure. Cracks often occur at twin-twin or twin-grain boundary intersections and propagate along twin bands. However, most crystal plasticity models for deformation twinning are based on a “pseudo-slip” approach and do not capture the localized deformation associated with the formation of each discrete twin band. The few exceptions are discrete twin models that involve very complex numerical algorithms and are often compromised in accuracy due to the numerical convergence. These factors make the discrete twin models hard to adopt. This paper proposes a modification to the conventional finite element weak form, to fully incorporate a twin-induced heterogeneous deformation that does not depend on the “pseudo-slip” assumption. The model starts by splitting the deformation gradient into elastic-slip-twinning components. The twin-induced deformation gradient component is computed separately by solving a microstructural evolution problem and then implemented into finite element weak form by constructing a global “twin-force” vector. The constitutive update (e.g., in the user-defined material subroutine, or UMAT, for ABAQUS) therefore avoids dealing with the twinning and recovers to the form of a regular slip-based crystal plasticity model. The results presented here indicate that the twin-induced strain localization and the associated stress-reversal phenomena near the twin band were naturally captured in the model, which was validated against an in-situ synchrotron X-ray micro-diffraction experiment.

36 MATERIALS SCIENCE↗