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NASA NTRS1989

A new family of stable elements for the Stokes problem based on a mixed Galerkin/least-squares finite element formulation

Adding to the classical Hellinger-Reissner formulation, a residual form of the equilibrium equation, a new Galerkin/least-squares finite element method is derived. It fits within the framework of a mixed finite element method and is stable for rather general combinations of stress and velocity interpolations, including equal-order discontinuous stress and continuous velocity interpolations which are unstable within the Galerkin approach. Error estimates are presented based on a generalization of the Babuska-Brezzi theory. Numerical results (not presented herein) have confirmed these estimates as well as the good accuracy and stability of the method.

Franca, Leopoldo P.↗

NASA NTRS1985

Prediction of Acoustical Response of Three-dimensional Cavities Using an Indirect Boundary Element Method

A technique which characterizes the acoustics of generalized cavities with the minimum model possible is developed. All boundary element methods have two advantages over finite element methods: (1) the models are smaller; and (2) the assumed variable behavior, inherent in the method to allow discretization, is harmonic rather than polynomial. Further, IBEM often requires one rather than two numerical boundary integrals as required by DBEM. Thus, a quadratic, isoparametric IBEM program was developed. The source distribution in this solution is continuous and quadratically variable rather than continuous and constant. The program was also formulated to include the additional capability of interior point sources and impedance boundary conditions. To test the quadratic, isoparametric IBEM program, several simple cavity enclosure problems where studied. Results are shown.

Bernhard, R. J.↗

NASA NTRS1992

Structures research

The main objective of the structures group is to provide quality aerospace research with the Center for Aerospace Research - A NASA Center for Excellence at North Carolina Agricultural and Technical State University. The group includes dedicated faculty and students who have a proven record in the area of structures, in particular space structures. The participating faculty developed accurate mathematical models and effective computational algorithms to characterize the flexibility parameters of joint dominated beam-truss structures. Both experimental and theoretical modelling has been applied to the dynamic mode shapes and mode frequencies for a large truss system. During the past few months, the above procedures has been applied to the hypersonic transport plane model. The plane structure has been modeled as a lumped mass system by Doctor Abu-Saba while Doctor Shen applied the transfer matrix method with a piecewise continuous Timoshenko tapered beam model. Results from both procedures compare favorably with those obtained using the finite element method. These two methods are more compact and require less computer time than the finite element method. The group intends to perform experiments on structural systems including the hypersonic plane model to verify the results from the theoretical models.

Abu-Saba, Elias↗

NASA NTRS1993

Solution of free-boundary problems using finite-element/Newton methods and locally refined grids - Application to analysis of solidification microstructure

A new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.

Tsiveriotis, K.↗

NASA NTRS1978

Trends in computerized structural analysis and synthesis; Proceedings of the Symposium, Washington, D.C., October 30-November 1, 1978

The subjects considered are related to future directions of structural applications and potential of new computing systems, advances and trends in data management and engineering software development, advances in applied mathematics and symbolic computing, computer-aided instruction and interactive computer graphics, nonlinear analysis, dynamic analysis and transient response, structural synthesis, structural analysis and design systems, advanced structural applications, supercomputers, numerical analysis, and trends in software systems. Attention is given to the reliability and optimality of the finite element method, computerized symbolic manipulation in structural mechanics, a standard computer graphics subroutine package, and a drag method as a finite element mesh generation scheme.

Noor, A. K.↗

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 NTRS1980

Stress-intensity factors for internal surface cracks in cylindrical pressure vessels

The paper presents stress-intensity factors for a wide range of semi-elliptical surface cracks on the inside of pressurized cylinders. The ratio of crack depth to crack length ranged from 0.2 to 1; the ratio of crack depth to wall thickness ranged from 0.2 to 0.8; and the ratio of wall thickness to vessel radius was 0.1 to to 0.25. The stress-intensity factors were calculated by a three-dimensional finite-element method using singularity elements along the crack front and linear-strain elements elsewhere. An equation for the stress-intensity factors was obtained which applies over a wide range of configuration parameters and was within about 5 percent of the present results. A comparison was also made between the results and other analyses of internal surface cracks in cylinders. The results from a boundary-integral equation method were in agreement and those from another finite-element method were in fair agreement (+ or - 8 percent) with the results.

Newman, J. C., Jr.↗

NASA NTRS2023

Modeling holes and voids in three dimensions using a single element within the Extended Finite Element framework

This paper introduces a highly efficient computational approach using the extended finite element method to model three-dimensional mechanics of holes and voids. The defect is encapsulated within a single element with a tailored exponential enrichment function that captures the behavior near the discontinuity boundary and can account for the pressure acting on the void’s surface through an equivalent nodal force vector. The method’s results were compared against analytical solutions and conventional finite element method results for problems involving single and multiple holes. The comparison demonstrated that the method provides accurate stress concentration estimates while adhering to computational constraints.

Adnan Shahriar↗

NASA NTRS1992

Coupling finite and boundary element methods for two-dimensional potential problems

A finite element-boundary element (FE-BE) coupling method based on a weighted residual variational method is presented for potential problems, governed by either the Laplace or the Poisson equations. In this method, a portion of the domain of interest is modeled by finite elements (FE) and the remainder of the region by boundary elements (BE). Because the BE fundamental solutions are valid for infinite domains, a procedure that limits the effects of the BE fundamental solution to a small region adjacent to the FE region, called the transition region (TR), is developed. This procedure involves a judicious choice of functions called the transition (T) functions that have unit values on the BE-TR interface and zero values on the FE-TR interface. The present FE-BE coupling algorithm is shown to be independent of the extent of the transition region and the choice of the transition functions. Therefore, transition regions that extend to only one layer of elements between FE and BE regions and the use of simple linear transition functions work well.

Krishnamurthy, T.↗

NASA NTRS2011

Modeling Electromagnetic Scattering From Complex Inhomogeneous Objects

This software innovation is designed to develop a mathematical formulation to estimate the electromagnetic scattering characteristics of complex, inhomogeneous objects using the finite-element-method (FEM) and method-of-moments (MoM) concepts, as well as to develop a FORTRAN code called FEMOM3DS (Finite Element Method and Method of Moments for 3-Dimensional Scattering), which will implement the steps that are described in the mathematical formulation. Very complex objects can be easily modeled, and the operator of the code is not required to know the details of electromagnetic theory to study electromagnetic scattering.

Deshpande, Manohar↗

NASA NTRS1993

Coupling finite and boundary element methods for 2-D elasticity problems

A finite element-boundary element (FE-BE) coupling method for two-dimensional elasticity problems is developed based on a weighted residual variational method in which a portion of the domain of interest is modeled by FEs and the remainder of the region by BEs. The performance of the FE-BE coupling method is demonstrated via applications to a simple 'patch test' problem and three-crack problems. The method passed the patch tests for various modeling configurations and yielded accurate strain energy release rates for the crack problems studied.

Krishnamurthy, T.↗

NASA NTRS1982

An exact zooming method for finite element analyses

An exact zooming technique which employs static condensation and exact structural reanalysis methods was developed. Successive application of static condensation reduces the system to one that is only associated with the degrees of freedom (dof) of the original model. Application of an exact static reanalysis technique permits the displacements at the dof of the original model that are contained in the zoomed portion of the structure to be obtained first. The response external to the zoom, as well as the response of additional dof within various levels of zooming, is computed. With the triangular factor of the stiffness matrix of the original system available, this approach involves only the solution of a system of equations of small order.

Hirai, I.↗

NASA NTRS1989

A study of composite plates with holes/inclusions

Delamination characterization of a plate with a hole/inclusion is presented. A closed form solution is developed to obtain stresses on the boundary of hole/inclusion in the plate. Once the location with the highest tangential stress is identified, a Finite Element Method analysis of a laminate under tensile loading is considered. The models are presented for the closed form and Finite Element Method. The examples are for (+ or - 35/0/90)s laminate of AS4/3501-6. Tensile, biaxial, and shear loads are considered.

Bense, Ronald↗

NASA NTRS1996

A Mixed Finite Volume Element Method for Flow Calculations in Porous Media

A key ingredient in the simulation of flow in porous media is the accurate determination of the velocities that drive the flow. The large scale irregularities of the geology, such as faults, fractures, and layers suggest the use of irregular grids in the simulation. Work has been done in applying the finite volume element (FVE) methodology as developed by McCormick in conjunction with mixed methods which were developed by Raviart and Thomas. The resulting mixed finite volume element discretization scheme has the potential to generate more accurate solutions than standard approaches. The focus of this paper is on a multilevel algorithm for solving the discrete mixed FVE equations. The algorithm uses a standard cell centered finite difference scheme as the 'coarse' level and the more accurate mixed FVE scheme as the 'fine' level. The algorithm appears to have potential as a fast solver for large size simulations of flow in porous media.

Jim E Jones↗

NASA NTRS1993

Automated Finite Element Modeling of Wing Structures for Shape Optimization

The displacement formulation of the finite element method is the most general and most widely used technique for structural analysis of airplane configurations. Modem structural synthesis techniques based on the finite element method have reached a certain maturity in recent years, and large airplane structures can now be optimized with respect to sizing type design variables for many load cases subject to a rich variety of constraints including stress, buckling, frequency, stiffness and aeroelastic constraints (Refs. 1-3). These structural synthesis capabilities use gradient based nonlinear programming techniques to search for improved designs. For these techniques to be practical a major improvement was required in computational cost of finite element analyses (needed repeatedly in the optimization process). Thus, associated with the progress in structural optimization, a new perspective of structural analysis has emerged, namely, structural analysis specialized for design optimization application, or.what is known as "design oriented structural analysis" (Ref. 4). This discipline includes approximation concepts and methods for obtaining behavior sensitivity information (Ref. 1), all needed to make the optimization of large structural systems (modeled by thousands of degrees of freedom and thousands of design variables) practical and cost effective.

Harvey, Michael Stephen↗

NASA NTRS1980

Failure analyses of composite bolted joints

The complex failure behavior exhibited by bolted joints of graphite epoxy (Hercules AS/3501) was investigated for the net tension, bearing and shearout failure modes using combined analytical and experimental techniques. Plane stress, linear elastic, finite element methods were employed to determine the two dimensional state of stress resulting from a loaded hole in a finite width, semiinfinite strip. The stresses predicted by the finite element method were verified by experiment to lend credence to the analysis. The influence of joint geometric parameters on the state of stress and resultant strength of the joint was also studied. The resulting functional relationships found to exist between bolted joint strength and the geometric parameters, were applied in the formulation of semiempirical strength models for the basic failure modes. A point stress failure criterion was successfully applied as the failure criterion for the net tension and shearout failure modes.

Wilson, D. W.↗

NASA NTRS1989

Semidiscrete Galerkin modelling of compressible viscous flow past a circular cone at incidence

A numerical study of the laminar and compressible boundary layer, about a circular cone in a supersonic free stream, is presented. It is thought that if accurate and efficient numerical schemes can be produced to solve the boundary layer equations, they can be joined to numerical codes that solve the inviscid outer flow. The combination of these numerical codes is competitive with the accurate, but computationally expensive, Navier-Stokes schemes. The primary goal is to develop a finite element method for the calculation of 3-D compressible laminar boundary layer about a yawed cone. The proposed method can, in principle, be extended to apply to the 3-D boundary layer of pointed bodies of arbitrary cross section. The 3-D boundary layer equations governing supersonic free stream flow about a cone are examined. The 3-D partial differential equations are reduced to 2-D integral equations by applying the Howarth, Mangler, Crocco transformations, a linear relation between viscosity, and a Blasius-type of similarity variable. This is equivalent to a Dorodnitsyn-type formulation. The reduced equations are independent of density and curvature effects, and resemble the weak form of the 2-D incompressible boundary layer equations in Cartesian coordinates. In addition the coordinate normal to the wall has been stretched, which reduces the gradients across the layer and provides high resolution near the surface. Utilizing the parabolic nature of the boundary layer equations, a finite element method is applied to the Dorodnitsyn formulation. The formulation is presented in a Petrov-Galerkin finite element form and discretized across the layer using linear interpolation functions. The finite element discretization yields a system of ordinary differential equations in the circumferential direction. The circumferential derivatives are solved by an implicit and noniterative finite difference marching scheme. Solutions are presented for a 15 deg half angle cone at angles of attack of 5 and 10 deg. The numerical solutions assume a laminar boundary layer with free stream Mach number of 7. Results include circumferential distribution of skin friction and surface heat transfer, and cross flow velocity distributions across the layer.

Meade, Andrew James, Jr.↗