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All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Application of digital pulse shaping by least squares method to ultrasonic signals in composites

For ultrasonic testing in composites, a signal processing tool is identified that can significantly enhance the sharpness of ultrasonic waveforms and provide clearer pictures of the nature of the material flaw. The technique artificially improves the resolution of the system to discrete events by pulse shaping the measured waveform based on the signal from a reference. The optimum pulse shape operator is determined from a least-squares method in the z-domain.

Kishoni, D.↗
NASA NTRS1979

A subroutine package for discrete estimation problems

In this paper we describe a well documented, compactly coded, storage efficient, thoroughly tested, and easy to use set of FORTRAN IV subroutines for use in Kalman filter or least-squares applications. The package contains both the UDU covariance factorization and the square root information filter algorithms developed at the Jet Propulsion Laboratory. Numerical reliability of the algorithms is a key feature of the package.

Bierman, G. J.↗
NASA NTRS1993

Accuracy of least-squares methods for the Navier-Stokes equations

Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations.

Bochev, Pavel B.↗
NASA NTRS1993

Accuracy of least-squares methods for the Navier-Stokes equations

Recently there has been substantial interest in least-squares finite element methods for velocity-vorticity-pressure formulations of the incompressible Navier-Stokes equations. The main cause for this interest is the fact that algorithms for the resulting discrete equations can be devised which require the solution of only symmetric, positive definite systems of algebraic equations. On the other hand, it is well-documented that methods using the vorticity as a primary variable often yield very poor approximations. Thus, here we study the accuracy of these methods through a series of computational experiments, and also comment on theoretical error estimates. It is found, despite the failure of standard methods for deriving error estimates, that computational evidence suggests that these methods are, at the least, nearly optimally accurate. Thus, in addition to the desirable matrix properties yielded by least-squares methods, one also obtains accurate approximations.

Bochev, Pavel B.↗
NASA NTRS1996

A Least-Squares Finite Element Method for Electromagnetic Scattering Problems

The least-squares finite element method (LSFEM) is applied to electromagnetic scattering and radar cross section (RCS) calculations. In contrast to most existing numerical approaches, in which divergence-free constraints are omitted, the LSFF-M directly incorporates two divergence equations in the discretization process. The importance of including the divergence equations is demonstrated by showing that otherwise spurious solutions with large divergence occur near the scatterers. The LSFEM is based on unstructured grids and possesses full flexibility in handling complex geometry and local refinement Moreover, the LSFEM does not require any special handling, such as upwinding, staggered grids, artificial dissipation, flux-differencing, etc. Implicit time discretization is used and the scheme is unconditionally stable. By using a matrix-free iterative method, the computational cost and memory requirement for the present scheme is competitive with other approaches. The accuracy of the LSFEM is verified by several benchmark test problems.

Wu, Jie↗
NASA NTRS1993

Development of a gridless CFD method

The objective of this paper is to stimulate ideas concerning alternative approaches to discretizing fluid flow problems, i.e., the so-called gridless approach. The gridless approach uses only clouds of points and does not require that the points be connected to form a grid as is necessary in conventional CFD algorithms. The governing equations are solved directly by performing local least-squares curve fits in each cloud of points and then analytically differentiating the resulting curve fits to approximate the derivatives. A discussion of these topics is presented in viewgraph form.

John T Batina↗
NASA NTRS1992

An improved plate theory of order (1,2) for thick composite laminates

A new (1,2)-order theory is proposed for the linear elasto-static analysis of laminated composite plates. The basic assumptions are those concerning the distribution through the laminate thickness of the displacements, transverse shear strains and the transverse normal stress, with these quantities regarded as some weighted averages of their exact elasticity theory representations. The displacement expansions are linear for the inplane components and quadratic for the transverse component, whereas the transverse shear strains and transverse normal stress are respectively quadratic and cubic through the thickness. The main distinguishing feature of the theory is that all strain and stress components are expressed in terms of the assumed displacements prior to the application of a variational principle. This is accomplished by an a priori least-square compatibility requirement for the transverse strains and by requiring exact stress boundary conditions at the top and bottom plate surfaces. Equations of equilibrium and associated Poisson boundary conditions are derived from the virtual work principle. It is shown that the theory is particularly suited for finite element discretization as it requires simple C(sup 0)- and C(sup -1)-continuous displacement interpolation fields. Analytic solutions for the problem of cylindrical bending are derived and compared with the exact elasticity solutions and those of our earlier (1,2)-order theory based on the assumed displacements and transverse strains.

Tessler, A.↗
NASA NTRS1996

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

Cai, Z.↗
NASA NTRS1993

Enhanced reduction of velocity data obtained during CETA flight experiment

A newly developed technique for enhanced data reduction provides an improved procedure that allows least squares minimization to become possible between data sets with an unequal number of data points. This technique was applied in the Crew and Equipment Translation Aid (CETA) experiment on the STS-37 Shuttle flight in April 1991 to obtain the velocity profile from the acceleration data. The new technique uses a least-squares method to estimate the initial conditions and calibration constants. These initial conditions are estimated by least-squares fitting the displacements indicated by the Hall-effect sensor data to the corresponding displacements obtained from integrating the acceleration data. The velocity and displacement profiles can then be recalculated from the corresponding acceleration data using the estimated parameters. This technique, which enables instantaneous velocities to be obtained from the test data instead of only average velocities at varying discrete times, offers more detailed velocity information, particularly during periods of large acceleration or deceleration.

Finley, Tom D.↗
NASA NTRS1993

Enhanced data reduction of the velocity data on CETA flight experiment

A newly developed technique for enhanced data reduction provides an improved procedure that allows least squares minimization to become possible between data sets with an unequal number of data points. This technique was applied in the Crew and Equipment Translation Aid (CETA) experiment on the STS-37 Shuttle flight in April 1991 to obtain the velocity profile from the acceleration data. The new technique uses a least-squares method to estimate the initial conditions and calibration constants. These initial conditions are estimated by least-squares fitting the displacements indicated by the Hall-effect sensor data to the corresponding displacements obtained from integrating the acceleration data. The velocity and displacement profiles can then be recalculated from the corresponding acceleration data using the estimated parameters. This technique, which enables instantaneous velocities to be obtained from the test data instead of only average velocities at varying discrete times, offers more detailed velocity information, particularly during periods of large acceleration or deceleration.

Finley, Tom D.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal estimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed.

Rodriguez, G.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal esimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed. While this paper focuses on theoretical developments, applications arising in the area of large structure static shape determination are contained in a closely related paper (Rodriguez and Scheid, 1982).

Rodriguez, G.↗
NASA NTRS1996

Three-Dimensional Simulations of Marangoni-Benard Convection in Small Containers by the Least-Squares Finite Element Method

This paper reports a numerical study of the Marangoni-Benard (MB) convection in a planar fluid layer. The least-squares finite element method (LSFEM) is employed to solve the three-dimensional Stokes equations and the energy equation. First, the governing equations are reduced to be first-order by introducing variables such as vorticity and heat fluxes. The resultant first-order system is then cast into a div-curl-grad formulation, and its ellipticity and permissible boundary conditions are readily proved. This numerical approach provides an equal-order discretization for velocity, pressure, vorticity, temperature, and heat conduction fluxes, and therefore can provide high fidelity solutions for the complex flow physics of the MB convection. Numerical results reported include the critical Marangoni numbers (M(sub ac)) for the onset of the convection in containers with various aspect ratios, and the planforms of supercritical MB flows. The numerical solutions compared favorably with the experimental results reported by Koschmieder et al..

Yu, Sheng-Tao↗
NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1991

Estimation of optimal Kalman filter gain from non-optimal filter residuals

This paper presents a novel method of estimating the optimal steady state Kalman filter gain of a linear discrete time-invariant system from a non-optimal Kalman filter residual sequence. The relation between the optimal residual sequence and a signal derived from the non-optimal residual sequence is described by a Moving Average (MA) model whose coefficients are expressed in terms of the state space parameters and the optimal steady state Kalman filter gain. In order to identify the MA model, a whitening filter of the derived signal, which corresponds to an AutoRegressive (AR) model of the signal, is first identified using the least-squares method. Then the inverse filter of the whitening filter, which corresponds to the MA model, is calculated. From the coefficients of the identified MA model, the optimal steady state Kalman filter gain can be obtained. Numerical example is provided to illustrate the feasibility of this approach.

Chen, Chung-Wen↗
DOE OSTI2026

3D seismic imaging of a fracture damage zone controlling reservoir compartmentalization at the Raft River EGS using multi-azimuth walkaway VSP

Accurate imaging of steeply dipping fracture zones in crystalline enhanced geothermal systems (EGS) is critical for constraining permeability architecture and guiding stimulation design. However, such structures remain poorly resolved by conventional surface seismic methods. We present a fully three-dimensional (3D) elastic-waveform inversion-migration workflow applied to multi-azimuth walkaway vertical seismic profiling (VSP) data acquired in a deviated borehole at the Raft River EGS. The workflow integrates first-arrival traveltime tomography, multi-scale elastic waveform inversion (EWI), and elastic least-squares reverse-time migration (ELSRTM) to recover high-resolution compressional-and shear-wave velocity models and to image structural discontinuities in the crystalline basement. The results reveal a laterally continuous low-velocity anomaly, with shear-wave velocity reductions of 25-30%, consistent with fractureinduced mechanical weakening. Two steeply dipping discontinuities bound a 50-80 m wide fracture damage zone. Independent constraints from microseismic clustering and geochemical compartmentalization corroborate the geometry and structural significance of this feature. Synthetic modeling further confirms that structures of this scale are resolvable (~30 m). These findings indicate that the Narrows structure is a distributed fracture damage zone rather than a discrete fault plane. By resolving fracture-zone geometry at the tens-of-meters scale using a single borehole, this workflow provides a practical and transferable approach for improving structural characterization, reducing uncertainty in permeability architecture, and supporting reservoir modeling and stimulation design in fractured crystalline EGS reservoirs.

58 GEOSCIENCES↗
NASA NTRS1993

Large-scale computation of incompressible viscous flow by least-squares finite element method

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Jiang, Bo-Nan↗
NASA NTRS1974

Sequential square root filtering and smoothing of discrete linear systems

A square root information filter/smoother is derived using recursive least-squares arguments. The combined filter/smoother algorithm has the following attributes: (1) it has a square root structure, which enhances numerical accuracy; (2) filter and smoother mechanizations are identical in form, facilitating implementation of the smoother; and (3) storage and computation requirements are modest compared with other smoothing algorithms. Partitioning the results to separate bias parameters provides further computational economies and reduction of storage requirements.

Bierman, G. J.↗