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The present study adapts the Z-matrix formulation using a mixed basis of finite elements and Gaussians. This is a energy-independent basis which allows flexible boundary conditions and is amenable to efficient algorithms for evaluating the necessary matrix elements with molecular targets.
Infinite-order generalized impedance boundary conditions (GIBCs) and generalized sheet transition conditions (GSTCs) for planar multilayer configurations are developed via the Taylor series expansion method. The conditions are derived in a matrix product form where each matrix corresponds to a specific layer. An overall composite boundary/transition condition is obtained by making finite-order approximations to the elements of each matrix for the cases of 'low-contrast' and 'high-contrast' material layers. The accuracy of the truncated boundary conditions is examined by comparing their implied reflection and transmission coefficients with the corresponding exact coefficients. Design curves are also given which relate the maximum order of the conditions required to simulate a coating or layer of specific thickness and contrast. Expressions are then derived for the reflection and transmission coefficients of the GIBC/GSTC sheets, and these are compared to exact coefficients to demonstrate the validity of the derived GIBCs/GSTCs.
A re-evaluation of the 3-D finite-element models and methods used to analyze surface crack at stress concentrations is presented. Previous finite-element models used by Raju and Newman for surface and corner cracks at holes were shown to have ill-shaped elements at the intersection of the hole and crack boundaries. These ill-shaped elements tended to make the model too stiff and, hence, gave lower stress-intensity factors near the hole-crack intersection than models without these elements. Improved models, without these ill-shaped elements, were developed for a surface crack at a circular hole and at a semi-circular edge notch. Stress-intensity factors were calculated by both the nodal-force and virtual-crack-closure methods. Both methods and different models gave essentially the same results. Comparisons made between the previously developed stress-intensity factor equations and the results from the improved models agreed well except for configurations with large notch-radii-to-plate-thickness ratios. Stress-intensity factors for a semi-elliptical surface crack located at the center of a semi-circular edge notch in a plate subjected to remote tensile loadings were calculated using the improved models. The ratio of crack depth to crack length ranged form 0.4 to 2; the ratio of crack depth to plate thickness ranged from 0.2 to 0.8; and the ratio of notch radius to the plate thickness ranged from 1 to 3. The models had about 15,000 degrees-of-freedom. Stress-intensity factors were calculated by using the nodal-force method.
The accurate prediction of wall-roughness effects in turbomachinery is becoming critical as turbine designers address airfoil surface quality and degradation concerns arising from the shift to advanced ceramic matrix composite (CMC) or additively-manufactured airfoils operating in higher temperature environments. In this paper, a recently developed computational capability for accurate and efficient scale-resolving simulations of turbomachinery is extended to analyze the boundary- layer separation and transition characteristics in a rough-wall low-pressure turbine (LPT) cascade. The computational capability is based on an entropy-stable discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in an efficient manner for a modern high performance computer architecture. Results from the scale-resolving simulations of both smooth and rough airfoil cascades are presented and compared to previous experiments and numerical simulations. The results show that the suction surface boundary layer undergoes laminar separation, transition, and turbulent reattachment for the smooth airfoil cascade, while in the presence of roughness the separation and transition behavior of the suction surface boundary layer is substantially modified. The differences between the smooth and rough airfoil cascades are then highlighted by a detailed analysis of their respective turbulent flow fields.
A concurrent multiscale modeling methodology that embeds a molecular dynamics (MD) region within a finite element (FEM) domain has been enhanced. The concurrent MD-FEM coupling methodology uses statistical averaging of the deformation of the atomistic MD domain to provide interface displacement boundary conditions to the surrounding continuum FEM region, which, in turn, generates interface reaction forces that are applied as piecewise constant traction boundary conditions to the MD domain. The enhancement is based on the addition of molecular dynamics-based cohesive zone model (CZM) elements near the MD-FEM interface. The CZM elements are a continuum interpretation of the traction-displacement relationships taken from MD simulations using Cohesive Zone Volume Elements (CZVE). The addition of CZM elements to the concurrent MD-FEM analysis provides a consistent set of atomistically-based cohesive properties within the finite element region near the growing crack. Another set of CZVEs are then used to extract revised CZM relationships from the enhanced embedded statistical coupling method (ESCM) simulation of an edge crack under uniaxial loading.
The topology-shape-size optimization of space structures is investigated through Kikuchi's homogenization method. The method starts from a 'design domain block,' which is a region of space into which the structure is to materialize. This domain is initially filled with a finite element mesh, typically regular. Force and displacement boundary conditions corresponding to applied loads and supports are applied at specific points in the domain. An optimal structure is to be 'carved out' of the design under two conditions: (1) a cost function is to be minimized, and (2) equality or inequality constraints are to be satisfied. The 'carving' process is accomplished by letting microstructure holes develop and grow in elements during the optimization process. These holes have a rectangular shape in two dimensions and a cubical shape in three dimensions, and may also rotate with respect to the reference axes. The properties of the perforated element are obtained through an homogenization procedure. Once a hole reaches the volume of the element, that element effectively disappears. The project has two phases. In the first phase the method was implemented as the combination of two computer programs: a finite element module, and an optimization driver. In the second part, focus is on the application of this technique to planetary structures. The finite element part of the method was programmed for the two-dimensional case using four-node quadrilateral elements to cover the design domain. An element homogenization technique different from that of Kikuchi and coworkers was implemented. The optimization driver is based on an augmented Lagrangian optimizer, with the volume constraint treated as a Courant penalty function. The optimizer has to be especially tuned to this type of optimization because the number of design variables can reach into the thousands. The driver is presently under development.
Component mode synthesis (CMS) is a method of dynamic analysis, for structures having a large number of degrees of freedom (DOF). These structures often required lengthy computer CPU time and large computer memory resources, if solved directly by the finite-element method (FEM). In CMS, the structure is divided into independent components in which the DOF are defined by a set of generalized coordinates defined by displacement shapes. The number of the generalized coordinates are much less than the original number of physical DOF, in the component. The displacement shapes are used to transform the component property matrices and any applied external loads, to a reduced system of coordinates. Reduced system property matrices are assembled, and any type of dynamic analysis is carried out in the reduced coordinate system. Any obtained results are back transformed to the original component coordinate systems. In all conventional methods of CMS, the mode shapes used for components are dynamic mode shapes, supplemented by static deflected shapes. Historically, all the dynamic mode shapes used in conventional CMS are the natural modes (eigenvectors) of components. A new method of CMS, namely the boundary flexibility vector method of CMS, is presented. The method provides for the incorporation of a set of static Ritz vectors, referred to as boundary flexibility vectors, as a replacement and/or supplement to conventional eigenvectors, as displacement shapes for components. The generation of these vectors does not require the solution of a costly eigenvalue problem, as in the case of natural modes in conventional CMS, and hence a substantial saving in CPU time can be achieved. The boundary flexibility vectors are generated from flexibility (or stiffness) properties of components. The formulation presented is for both free and fixed-interface components, and for both the free and forced vibration problems. Free and forced vibration numerical examples are presented to verify the accuracy of the method and the saving in CPU time. Compared to conventional methods of CMS, the results indicate that by using the new method, more accurate results can be obtained with a substantial saving in CPU time.
This study deals with the effect of assumed tow architecture on the elastic material properties and stress distributions of plain weave woven composites. Specifically, the examination of how a cross-section is assumed to sweep-out the tows of the composite is examined in great detail. The two methods studied are extrusion and translation. This effect is also examined to determine how sensitive this assumption is to changes in waviness ratio. 3D finite elements were used to study a T300/Epoxy plain weave composite with symmetrically stacked mats. 1/32nd of the unit cell is shown to be adequate for analysis of this type of configuration with the appropriate set of boundary conditions. At low waviness, results indicate that for prediction of elastic properties, either method is adequate. At high waviness, certain elastic properties become more sensitive to the method used. Stress distributions at high waviness ratio are shown to vary greatly depending on the type of loading applied. At low waviness, both methods produce similar results.
Geometric non-linearities for large amplitude free and forced vibrations of circular plates are investigated. In-plane displacement and in-plane inertia are included in the formulation. The finite element method is used. An harmonic force matrix for non-linear forced vibration analysis is introduced and derived. Various out-of-plane and in-plane boundary conditions are considered. The relations of amplitude and frequency ratio for different boundary conditions and various load conditions are presented.
A method for the fabrication of hot-wire sensors measuring 0.6 micron in diameter and 200 microns in length is described. This method allows accurate control of sensing-element orientation, thus making it possible to produce multisensor probes. Test results obtained with a miniature sensor fabricated by this method, for the outer region of a turbulent boundary layer developing in a zero presure gradient were compared to measurements from a conventional-sized probe made in the same flow, demonstrating that, because of their improved spatial resolution, subminiature sensors can provide more accurate measurements in boundary layers than do the conventional probes.
A two-step Taylor-Galerkin finite element method with Lapidus' artificial viscosity scheme is applied to several test cases for internal compressible inviscid flow problems. Investigations for the effect of supersonic/subsonic inlet and outlet boundary conditions on computational results are particularly emphasized.
A program for the generation of unstructured grids over complex configurations, VGRID3D, is described. The grid elements (triangles on the surfaces and tetrahedra in the field) are generated starting from the surface boundaries towards the interior of the computational domain using the Advancing Front Method.
The feasibility of changing the bolt shape from circular to elliptical in order to increase the joint strength was investigated. An analytical method using a cosine series to represent the bearing stress was derived, and a boundary collocation method was used to determine the unknown coefficients of this cosine series. Stresses at the hole edge predicted by the analytical method agreed very well with finite element solutions. Failure analyses of joints in two different laminates were performed, and each laminate exhibited a different joint failure mode. Results demonstrate that the joint strength of both laminates can be improved substantially by changing the bolt shape to elliptical. The joint that failed in a bearing mode showed a greater strength increase compared to the joint that failed in a shearing mode.
The dynamics of a drop of a Newtonian liquid that is pendant from or sessile on a solid rod that is forced to undergo time-periodic oscillations along its axis is studied theoretically. The free boundary problem governing the time evolution of the shape of the drop and the flow field inside it is solved by a method of lines using a finite element algorithm incorporating an adaptive mesh. When the forcing amplitude is small, the drop approaches a limit cycle at large times and undergoes steady oscillations thereafter. However, drop breakup is the consequence if the forcing amplitude exceeds a critical value. Over a wide range of amplitudes above this critical value, drop ejection from the rod occurs during the second oscillation period from the commencement of rod motion. Remarkably, the shape of the interface at breakup and the volume of the primary drop formed are insensitive to changes in forcing amplitude. The interface shape at times close to and at breakup is a multi-valued function of distance measured along the rod axis and hence cannot be described by recently popularized one-dimensional approximations. The computations show that drop ejection occurs without the formation of a long neck. Therefore, this method of drop formation holds promise of preventing formation of undesirable satellite droplets.
A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions and the pressure was interpolated using linear shape functions. For the two dimensional case, the pressure is defined on a triangular element which is contained inside the complete biquadratic element for velocity variables; and for the three dimensional case, the pressure is defined on a tetrahedral element which is again contained inside the complete tri-quadratic element. Thus the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow for Reynolds number of 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorable with those of the finite difference methods as well as experimental data available. A finite elememt computer program for incompressible, laminar flows is presented.
Viscous flow over discrete or distributed surface roughness has great implications for hypersonic flight due to aerothermodynamic considerations related to laminar-turbulent transition. Current prediction capability is greatly hampered by the limited knowledge base for such flows. To help fill that gap, numerical computations are used to investigate the intricate flow physics involved. An unstructured mesh, compressible Navier-Stokes code based on the space-time conservation element, solution element (CESE) method is used to perform time-accurate Navier-Stokes calculations for two roughness shapes investigated in wind tunnel experiments at NASA Langley Research Center. It was found through 2D parametric study that at subcritical Reynolds numbers of the boundary layers, absolute instability resulting in vortex shedding downstream, is likely to weaken at supersonic free-stream conditions. On the other hand, convective instability may be the dominant mechanism for supersonic boundary layers. Three-dimensional calculations for a rectangular or cylindrical roughness element at post-shock Mach numbers of 4.1 and 6.5 also confirm that no self-sustained vortex generation is present.
Fatigue cracks are a major problem in designing structures subjected to cyclic loading. Cracks frequently occur in structures such as aircraft and spacecraft. The inspection intervals of many aircraft structures are based on crack-propagation lives. Therefore, improved prediction of propagation lives under flight-load conditions (variable-amplitude loading) are needed to provide more realistic design criteria for these structures. The main thrust was to develop a three-dimensional, nonlinear, elastic-plastic, finite element program capable of extending a crack and changing boundary conditions for the model under consideration. The finite-element model is composed of 8-noded (linear-strain) isoparametric elements. In the analysis, the material is assumed to be elastic-perfectly plastic. The cycle stress-strain curve for the material is shown Zienkiewicz's initial-stress method, von Mises's yield criterion, and Drucker's normality condition under small-strain assumptions are used to account for plasticity. The three-dimensional analysis is capable of extending the crack and changing boundary conditions under cyclic loading.
The internal fluid mechanics research program in inlets, ducts, and nozzles is described. The program consists of a balanced effort between the development of computational tools and the conduct of experimental research. The experiments are designed to better understand the fluid flow physics, to develop new or improved flow models, and to provide benchmark quality data sets for validation of the computational methods. The inlet, duct, and nozzle research program is described according to three major classifications of flow phenomena: highly three-dimensional flow fields; shock-boundary layer interactions; and shear layer control. Specific examples of current and future elements of the research program are described for each of these phenomena. In particular, the highly three-dimensional flow field phenomena is highlighted by describing the computational and experiemental research program in transition ducts having a round-to-rectangular area variation. In the case of shock-boundary layer interactions, the specific details of research for normal shock-boundary layer interactions are described. For shear layer control research in vortex generators and the use of aerodynamic excitation for enhancement of the jet mixing process are described. Future research in inlets, ducts, and nozzles will include more emphasis on three-dimensional full Navier-Stokes methods and corresponding experiments designed to concentrate on the appropriate three-dimensional fluid flow physics.