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NASA NTRS1974

Large deflection elastic-plastic dynamic response of stiffened shells of revolution

This paper presents the formulation and check-out problems for a computer code DYNAPLAS, which analyzes the large deflection elastic-plastic dynamic response of stiffened shells of revolution. The formulation for spacial discretization is by the finite element method with finite differences being used for the evaluation of the pseudo forces due to material and geometric nonlinearities. Time integration is by the Houbolt method or central differences. The stiffeners may be due to concentrated or distributed eccentric rings and spring supports at arbitrary angles around the circumference of the elements. Check-out problems include the comparison of solutions from DYNAPLAS with experimental and other computer solutions for rings and conical and cylindrical shells. A hypothetical submarine including stiffeners and missile tube is studied under a combination of hydrostatic and dynamically applied asymmetrical pressure loadings.

Stricklin, J. A.↗

NASA NTRS1987

Aeroelastic modeling of swept tip rotor blades using finite elements

A special finite element for the modeling of a swept tip rotor blade is derived. The swept tip blade undergoes moderate deflections in flap, lag and torsion. The nonlinear, partial differential equations of motion are discretized using a Galerkin finite element method. Tip sweep introduces flap-torsion and lag-axial couplings, and may lead to aeroelastic instabilities associated with frequency coalescence. When frequency coalescence does not occur, sweep is usually stabilizing. A comparison of the approximate model of a swept tip blade, using a blade with straight elastic axis and offsets of aerodynamic centers of gravity, with the exact swept tip model developed in this paper indicates that the approximate model can produce inaccurate results for the case of hingeless rotor blades.

Celi, R.↗

NASA NTRS1988

Aeroelastic effects on stability and control of hingeless rotor helicopters

The combined effect of torsional flexibility and offset of the aerodynamic center relative to the elastic axis on the stability and control of a hingeless rotor helicopter in forward flight is studied in this paper. The aeroelastic model of the blade is based on fully coupled flap-lag-torsion equations of motion, which include kinematic nonlinearities due to moderate deflections. The equations are discretized using a finite element Galerkin method in space, and a classical Galerkin method in time. The vehicle trim calculations are coupled to the blade aeroelastic response calculations. Quasilinearization is used to compute aeroelastic stability. Reducing torsional stiffness and moving the elastic axis ahead of the aerodynamic center strongly stabilizes the phugoid mode without reducing control sensitivity. Aeroelastic stability calculations must accompany flight dynamics calculations to prevent aeroelastic instabilities that might easily go undetected.

Celi, Roberto↗

NASA NTRS1996

Simulation of Two-Fluid Flows by the Least-Squares Finite Element Method Using a Continuum Surface Tension Model

In this paper a numerical procedure for simulating two-fluid flows is presented. This procedure is based on the Volume of Fluid (VOF) method proposed by Hirt and Nichols and the continuum surface force (CSF) model developed by Brackbill, et al. In the VOF method fluids of different properties are identified through the use of a continuous field variable (color function). The color function assigns a unique constant (color) to each fluid. The interfaces between different fluids are distinct due to sharp gradients of the color function. The evolution of the interfaces is captured by solving the convective equation of the color function. The CSF model is used as a means to treat surface tension effect at the interfaces. Here a modified version of the CSF model, proposed by Jacqmin, is used to calculate the tension force. In the modified version, the force term is obtained by calculating the divergence of a stress tensor defined by the gradient of the color function. In its analytical form, this stress formulation is equivalent to the original CSF model. Numerically, however, the use of the stress formulation has some advantages over the original CSF model, as it bypasses the difficulty in approximating the curvatures of the interfaces. The least-squares finite element method (LSFEM) is used to discretize the governing equation systems. The LSFEM has proven to be effective in solving incompressible Navier-Stokes equations and pure convection equations, making it an ideal candidate for the present applications. The LSFEM handles all the equations in a unified manner without any additional special treatment such as upwinding or artificial dissipation. Various bench mark tests have been carried out for both two dimensional planar and axisymmetric flows, including a dam breaking, oscillating and stationary bubbles and a conical liquid sheet in a pressure swirl atomizer.

Wu, Jie↗

NASA NTRS2001

Multifunctional Collaborative Modeling and Analysis Methods in Engineering Science

Engineers are challenged to produce better designs in less time and for less cost. Hence, to investigate novel and revolutionary design concepts, accurate, high-fidelity results must be assimilated rapidly into the design, analysis, and simulation process. This assimilation should consider diverse mathematical modeling and multi-discipline interactions necessitated by concepts exploiting advanced materials and structures. Integrated high-fidelity methods with diverse engineering applications provide the enabling technologies to assimilate these high-fidelity, multi-disciplinary results rapidly at an early stage in the design. These integrated methods must be multifunctional, collaborative, and applicable to the general field of engineering science and mechanics. Multifunctional methodologies and analysis procedures are formulated for interfacing diverse subdomain idealizations including multi-fidelity modeling methods and multi-discipline analysis methods. These methods, based on the method of weighted residuals, ensure accurate compatibility of primary and secondary variables across the subdomain interfaces. Methods are developed using diverse mathematical modeling (i.e., finite difference and finite element methods) and multi-fidelity modeling among the subdomains. Several benchmark scalar-field and vector-field problems in engineering science are presented with extensions to multidisciplinary problems. Results for all problems presented are in overall good agreement with the exact analytical solution or the reference numerical solution. Based on the results, the integrated modeling approach using the finite element method for multi-fidelity discretization among the subdomains is identified as most robust. The multiple-method approach is advantageous when interfacing diverse disciplines in which each of the method's strengths are utilized. The multifunctional methodology presented provides an effective mechanism by which domains with diverse idealizations are interfaced. This capability rapidly provides the high-fidelity results needed in the early design phase. Moreover, the capability is applicable to the general field of engineering science and mechanics. Hence, it provides a collaborative capability that accounts for interactions among engineering analysis methods.

Ransom, Jonathan B.↗

NASA NTRS2001

On Multifunctional Collaborative Methods in Engineering Science

Multifunctional methodologies and analysis procedures are formulated for interfacing diverse subdomain idealizations including multi-fidelity modeling methods and multi-discipline analysis methods. These methods, based on the method of weighted residuals, ensure accurate compatibility of primary and secondary variables across the subdomain interfaces. Methods are developed using diverse mathematical modeling (i.e., finite difference and finite element methods) and multi-fidelity modeling among the subdomains. Several benchmark scalar-field and vector-field problems in engineering science are presented with extensions to multidisciplinary problems. Results for all problems presented are in overall good agreement with the exact analytical solution or the reference numerical solution. Based on the results, the integrated modeling approach using the finite element method for multi-fidelity discretization among the subdomains is identified as most robust. The multiple method approach is advantageous when interfacing diverse disciplines in which each of the method's strengths are utilized.

Ransom, Jonathan B.↗

NASA NTRS1993

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

Scott, James R.↗

DOE OSTI2020

Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting

This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. Finally, the convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite-volume discretization for the multiphase flow equations coupled with a finite-element scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the embedded discrete fracture model (EDFM) and the embedded finite element method (EFEM) for the flow and the mechanics, respectively. This nonconforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finite volumes that exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite-element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an extended finite element approach. One key advantage of EFEM is the element-based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

58 GEOSCIENCES↗

NASA NTRS2018

Geometry and Joint Systems for Lattice-Based Reconfigurable Space Structures

We describe analytical methods for the design of the discrete elements of ultralight lattice structures. This modular, building block strategy allows for relatively simple element manufacturing, as well as relatively simple robotic assembly of low mass density structures on orbit, with potential for disassembly and reassembly into highly varying and large structures. This method also results in a structure that is easily navigable by relatively small mobile robots. The geometry of the cell can allow for high packing efficiency to minimize wasted payload volume while maximizing structural performance and constructability. We describe the effect of geometry choices on the final system mechanical properties, manufacturability of the components, and automated robotic constructability of a final system. Geometry choices considered include building block complexity, symmetry of the unit cell, and effects of vertex, edge, and face connectivity of the unit cell. Mechanical properties considered include strength scaling, modulus scaling, and structural performance of the joint, including proof load, shear load, mass, and loading area; as well as validation and verification opportunities. Manufacturability metrics include cost and time, manufacturing method (COTS versus custom), and tolerances required. Automated constructability metrics include local effects of loads imparted to the structure by the robot and assembly complexity, encompassing the ability of the robot to clamp and number of placement motions needed for assembly.

Ochalek, Megan E.↗

NASA NTRS2019

Geometry Systems for Lattice-Based Reconfigurable Space Structures

We describe analytical methods for the design of the discrete elements of ultralight lattice structures. This modular, building block strategy allows for relatively simple element manufacturing, as well as relatively simple robotic assembly of low mass density structures on orbit, with potential for disassembly and reassembly into highly varying and large structures. This method also results in a structure that is easily navigable by relatively small mobile robots. The geometry of the cell can allow for high packing efficiency to minimize wasted payload volume while maximizing structural performance and constructability. We describe the effect of geometry choices on the final system mechanical properties and automated robotic constructability of a final system. Geometric properties considered include number of attachments per voxel, number of attachments per coefficient of volume, and effects of vertex, edge, and face connectivity of the unit cell. Mechanical properties considered include strength scaling, modulus scaling, and packing efficiency of the lattice. Automated constructibility metrics include volume allowance for an end-effector, strut clearance angle for an end-effector, and packing efficiency. These metrics were applied to six lattice unit cell geometries: cube, cuboctahedron, octahedron, octet, rhombic dodecahedron, and truncated octahedron. A case study is presented to determine the most suitable lattice system for a specific set of strength and modulus scaling requirements while optimizing for ease of robotic assembly.

lattice geometry↗

NASA NTRSDate not supplied

Geometry Systems for Lattice-Based Reconfigurable Space Structures

We describe analytic methods for the design of the discrete elements of ultralight lattice structures. This modular building block strategy allows for relatively simple element manufacturing as well as relatively simple robotic assembly of low mass-density structures on orbit, with potential for disassembly and reassembly into highly varying and large structures. This method also results in a structure that is easily navigable by relatively small, mobile robots. The geometry of the cell can allow for high packing efficiency to minimize wasted payload volume while maximizing structural performance and constructability. We describe the effect of geometry choices on the mechanical properties and automated robotic constructability of a final system. Geometric properties considered include number of attachments per voxel, number of attachments per coefficient of volume, and effects of vertex, edge, and face connectivity of the unit cell. Mechanical properties considered include strength scaling, modulus scaling, and packing efficiency of the lattice. Automated constructibility metrics include volume allowance for an end-effector, strut clearance angle for an end-effector, and packing efficiency. These metrics were applied to six lattice unit cell geometries: cube, cuboctahedron, octahedron, octet, rhombic dodecahedron, and truncated octahedron. A case study is presented to determine the most suitable lattice system for a specific set of strength and modulus scaling requirements while optimizing for ease of robotic assembly.

digital materials↗

NASA NTRS1993

Compressible turbulent flow simulation with a multigrid multiblock method

We describe a multigrid multiblock method for compressible turbulent flow simulations and present results obtained from calculations on a two-element airfoil. A vertex-based spatial discretization method and explicit multistage Runge-Kutta time-stepping are used. The slow convergence of a single grid method makes the multigrid method, which yields a speed factor of about 20, indispensable. The numerical predictions are in good agreement with experimental results. It is shown that the convergence of the multigrid process depends considerably on the ordering of the various loops. If the block loop is put inside the stage loop, the process converges more rapidly than if the block loop is situated outside the stage loop when a three-stage Runge-Kutta method is used. If a five-stage scheme is used, the process does not converge in the latter block ordering. Finally, the process based on the five-stage method is about 60 percent more efficient than with the three-stage method, if the block loop is inside the stage loop.

Kuerten, Hans↗

DOE OSTI2020

A hybrid calibration approach to Hertz-type contact parameters for discrete element models

This study aims at providing a hybrid calibration framework to estimate Hertz-type contact parameters (particle-scale shear modulus and Poisson ratio) for both two-dimensional and three-dimensional discrete element modelling (DEM). On the basis of statistically isotropic granular packings, a set of analytical formulae between macroscopic material parameters (Young modulus and Poisson ratio) and particle-scale Hertz-type contact parameters for granular systems are derived under small-strain isotropic stress conditions. However, the derived analytical solutions are only estimated values for general models. By viewing each DEM modelling as an implicit mathematical function taking the particle-level parameters as independent variables and employing the derived analytical solutions as the initial input parameters, an automatic iterative scheme is proposed to obtain the calibrated parameters with higher accuracies. Considering highly nonlinear features and discontinuities of the macro-micro relationship in Hertz-based discrete element models, the adaptive moment estimation algorithm is adopted in this study because of its capacity of dealing with noise gradients of cost functions. Here, the proposed method is validated with several numerical cases including randomly distributed monodisperse and polydisperse packings. Noticeable improvements in terms of calibration efficiency and accuracy have been made.

Constitutive law↗

NASA NTRS1986

Iterative methods for elliptic finite element equations on general meshes

Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.

Nicolaides, R. A.↗

NASA NTRS1990

A comparison of internal energy calculation methods for diatomic molecules

Various methods of calculating the internal energy of diatomic molecules are studied. An accurate and efficient method for computing the eigenvalues of the vibrational Schroedinger equation for an arbitrary potential is developed. The method is based on a finite-element discretization using the cubic Lobatto element. A combination of spectrum slicing and the Laguerre algorithm is used to solve for the eigenvalues. A simple method to compute the quasi-bound states is presented. For N2 molecules, all vibrational-rotational states of eleven available electronic potentials are computed, and summed to obtain the exact internal energy function with temperature. The total computation required 314 seconds of CPU-time on NASA's Cray 2 computer. Various approximate models are discussed and compared with the exact numerical simulation. It is shown that the splitting of the macroscopic internal energy into separate electronic, rotational, and vibrational energies is not justified at high temperatures.

Liu, Yen↗

NASA NTRS1984

On accuracy conditions for the numerical computation of waves

The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models.

Bayliss, A.↗

NASA NTRS1985

On accuracy conditions for the numerical computation of waves

Bayliss, A.↗