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At least 253 records · Page 14

Iterative methods for mixed finite element equations

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Nakazawa, S.↗

Unintended Consequences: How Qualification Constrains Innovation

The development and implementation of new materials and manufacturing processes for aerospace application is often hindered by the high cost and long time span associated with current qualification procedures. The data requirements necessary for material and process qualification are extensive and often require millions of dollars and multiple years to complete. Furthermore, these qualification data can become obsolete for even minor changes to the processing route. This burden is a serious impediment to the pursuit of revolutionary new materials and more affordable processing methods for air vehicle structures. The application of integrated computational materials engineering methods to this problem can help to reduce the barriers to rapid insertion of new materials and processes. By establishing predictive capability for the development of microstructural features in relation to processing and relating this to critical property characteristics, a streamlined approach to qualification is possible. This paper critically examines the advantages and challenges to a modeling-assisted qualification approach for aerospace structural materials. An example of how this approach might apply towards the emerging field of additive manufacturing is discussed in detail.

Brice, Craig A.↗

On the identification of a harmonic force on a viscoelastic plate from response data

The problem of determining the force acting on a structure from measurements of the response of the structure to the force is an inverse problem. Presented is a method for determining the location, magnitude, and phase of a harmonic point force acting on a simply-supported classical viscoelastic rectangular plate from a number of displacement readings at discrete points on the plate. Presented also is a demonstration of the robustness of the solution technique to the effects of measurement noise as well as a means by which problems involving more general structural and loading configurations may be solved.

D'Cruz, J.↗

Methods for structural design at elevated temperatures

A procedure which can be used to design elevated temperature structures is discussed. The desired goal is to have the same confidence in the structural integrity at elevated temperature as the factor of safety gives on mechanical loads at room temperature. Methods of design and analysis for creep, creep rupture, and creep buckling are presented. Example problems are included to illustrate the analytical methods. Creep data for some common structural materials are presented. Appendix B is description, user's manual, and listing for the creep analysis program. The program predicts time to a given creep or to creep rupture for a material subjected to a specified stress-temperature-time spectrum. Fatigue at elevated temperature is discussed. Methods of analysis for high stress-low cycle fatigue, fatigue below the creep range, and fatigue in the creep range are included. The interaction of thermal fatigue and mechanical loads is considered, and a detailed approach to fatigue analysis is given for structures operating below the creep range.

Ellison, A. M.↗

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within “chemical accuracy” (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob’s ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 − N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Chemistry↗

Implementation of the Lanczos method for structural vibration analysis on a parallel computer

The use of the Lanczos method in a parallel environment is investigated by implementing the algorithm for structural vibration problems on a parallel computer. It is shown that introducing shifts, assigning each processor a different region in the eigenvalue spectrum, and implementing the Lanczos method in parallel is an effective strategy for speeding up calculations. Test problem results include examples of the 'Lanczos phenomenon' where roundoff error in the vector orthogonalization can result in spurious eigenvalues which must be identified and discarded. The calculation strategy described here permits straightforward determination of these spurious eigenvalues. It is demonstrated that significant speedups in calculation time can be realized over traditional sequential methods.

Bostic, S. W.↗

Off-Axis Nulling Transfer Function Measurement: A First Assessment

We want to study a polychromatic inverse problem method with nulling interferometers to obtain information on the structures of the exozodiacal light. For this reason, during the first semester of 2013, thanks to the support of the consortium PERSEE, we launched a campaign of laboratory measurements with the nulling interferometric test bench PERSEE, operating with 9 spectral channels between J and K bands. Our objective is to characterise the transfer function, i.e. the map of the null as a function of wavelength for an off-axis source, the null being optimised on the central source or on the source photocenter. We were able to reach on-axis null depths better than 10(exp −4). This work is part of a broader project aiming at creating a simulator of a nulling interferometer in which typical noises of a real instrument are introduced. We present here our first results.

interferometric test bench PERSEE↗

Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids—A case study in diamond

While Diffusion Monte Carlo (DMC) is in principle an exact stochastic method for ab initio electronic structure calculations, in practice, the fermionic sign problem necessitates the use of the fixed-node approximation and trial wavefunctions with approximate nodes (or zeros). This approximation introduces a variational error in the energy that potentially can be tested and systematically improved. Here, we present a computational method that produces trial wavefunctions with systematically improvable nodes for DMC calculations of periodic solids. These trial wavefunctions are efficiently generated with the configuration interaction using a perturbative selection made iteratively (CIPSI) method. A simple protocol in which both exact and approximate results for finite supercells are used to extrapolate to the thermodynamic limit is introduced. This approach is illustrated in the case of the carbon diamond using Slater–Jastrow trial wavefunctions including up to one million Slater determinants. Fixed-node DMC energies obtained with such large expansions are much improved, and the fixed-node error is found to decrease monotonically and smoothly as a function of the number of determinants in the trial wavefunction, a property opening the way to a better control of this error. The cohesive energy extrapolated to the thermodynamic limit is in close agreement with the estimated experimental value. Interestingly, this is also the case at the single-determinant level, thus, indicating a very good error cancellation in carbon diamond between the bulk and atomic total fixed-node energies when using single-determinant nodes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Dynamic transformation method

Eigenvalue problem associated with modal-synthesis vibration analysis of complex structures requires simplifying assumptions for solution. Computer program, Dynamic-transformation Adapted to Modal-synthesis Using Stiffness-coupling, improves computational economy for vibration analysis of complex structures while still considering substructure modes.

Admire, J. R.↗

Run-time scheduling and execution of loops on message passing machines

Sparse system solvers and general purpose codes for solving partial differential equations are examples of the many types of problems whose irregularity can result in poor performance on distributed memory machines. Often, the data structures used in these problems are very flexible. Crucial details concerning loop dependences are encoded in these structures rather than being explicitly represented in the program. Good methods for parallelizing and partitioning these types of problems require assignment of computations in rather arbitrary ways. Naive implementations of programs on distributed memory machines requiring general loop partitions can be extremely inefficient. Instead, the scheduling mechanism needs to capture the data reference patterns of the loops in order to partition the problem. First, the indices assigned to each processor must be locally numbered. Next, it is necessary to precompute what information is needed by each processor at various points in the computation. The precomputed information is then used to generate an execution template designed to carry out the computation, communication, and partitioning of data, in an optimized manner. The design is presented for a general preprocessor and schedule executer, the structures of which do not vary, even though the details of the computation and of the type of information are problem dependent.

Crowley, Kay↗

Run-time scheduling and execution of loops on message passing machines

Sparse system solvers and general purpose codes for solving partial differential equations are examples of the many types of problems whose irregularity can result in poor performance on distributed memory machines. Often, the data structures used in these problems are very flexible. Crucial details concerning loop dependences are encoded in these structures rather than being explicitly represented in the program. Good methods for parallelizing and partitioning these types of problems require assignment of computations in rather arbitrary ways. Naive implementations of programs on distributed memory machines requiring general loop partitions can be extremely inefficient. Instead, the scheduling mechanism needs to capture the data reference patterns of the loops in order to partition the problem. First, the indices assigned to each processor must be locally numbered. Next, it is necessary to precompute what information is needed by each processor at various points in the computation. The precomputed information is then used to generate an execution template designed to carry out the computation, communication, and partitioning of data, in an optimized manner. The design is presented for a general preprocessor and schedule executer, the structures of which do not vary, even though the details of the computation and of the type of information are problem dependent.

Saltz, Joel↗

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of φ $-$functions of matrices via quadrature

Here, in this article, we propose an algorithm for approximating the action of φ $-$ functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ $-$ functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

97 MATHEMATICS AND COMPUTING↗

CP decomposition for tensors via alternating least squares with QR decomposition

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

97 MATHEMATICS AND COMPUTING↗

Eigenvalue and eigenvector sensitivity and approximate analysis for repeated eigenvalue problems

A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector approximate analysis in the presence of repeated eigenvalues is presented. The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and approximate analysis.

Hou, Gene J. W.↗

Variational approach to probabilistic finite elements

Probabilistic finite element method (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties, and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗

Variational approach to probabilistic finite elements

Probabilistic finite element methods (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗

PowerMappeR: Power-Optimized Mapping of SNNs onto ReRAM Crossbars coupled via Packet-Switched NoCs

Many recent efforts in developing hardware-accelerated spiking neural networks (SNNs) are characterized by deep co-design between algorithms, architectures, and devices. Architectural advances overcome device constraints by coupling together many small resistive-RAM (ReRAM) crossbars via a network-on-chip (NoC) for neuromorphic component operation. Concurrently, improved SNN training methods increase accuracy and structural sparsity in networks despite growing problem sizes. Finally, compilers leverage these attributes to minimize area and inter-crossbar communication while mapping large SNNs to sophisticated architectures. However, for compiler-driven co-design to realize increasingly complex and profitable optimizations, a compile-time view of power consumption is critical. We present PowerMappeR to express and optimize over mapping-, architecture-, and device-specific power consumption information. By modeling the dynamic power of well-established components, we develop an integer linear programming (ILP)-based, encoding-agnostic, parametric power estimation model. Using this model, we demonstrate practical improvements in area and inter-crossbar communication by 0%–9.5% and 1.4%–5.1%, respectively. We also limit hotspot formation during optimization, achieving comparable or better results in targeted metrics with up to 96.4%–97.1% restriction of hotspot magnitude. Finally, we introduce profile-guided formulations to reduce worst-case and expected-case hotspot magnitude by 40.7%–69.5% and 40.6%–56.3%, respectively. Optimizing worst-case hotspot magnitude incidentally improves expected-case magnitude by 10.85%–33.45%. Reciprocally, optimizing expected-case magnitude incidentally improves worst-case magnitude by 4.33%–39.87%. Validation against hardware simulators confirms that PowerMappeR can decrease dynamic power consumption by 12.6%–27.3%.

Pohl, Devin [ORNL] (ORCID:0009000040149027)↗

Some trends in aircraft design: Structures

Trends and programs currently underway on the national scene to improve the structural interface in the aircraft design process are discussed. The National Aeronautics and Space Administration shares a partnership with the educational and industrial community in the development of the tools, the criteria, and the data base essential to produce high-performance and cost-effective vehicles. Several thrusts to build the technology in materials, structural concepts, analytical programs, and integrated design procedures essential for performing the trade-offs required to fashion competitive vehicles are presented. The application of advanced fibrous composites, improved methods for structural analysis, and continued attention to important peripheral problems of aeroelastic and thermal stability are among the topics considered.

Brooks, G. W.↗