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A conforming mixed finite element method for the stationary Navier-Stokes equations with inhomogeneous boundary conditions is considered. Estimates for the error in the velocity and pressure approximations are derived. A computational example is provided to illustrate the implementation of the method.
Much of the computational effort of the finite element process involves the solution of a system of linear equations. The coefficient matrix of this system, known as the global stiffness matrix, is symmetric, positive definite, and generally sparse. An important technique for reducing the time required to solve this system is substructuring or matrix partitioning. Substructuring is based on the idea of dividing a structure into pieces, each of which can then be analyzed relatively indepenently. As a result of this division, each point in the finite element discretization is either interior to a substructure or on a boundary between substructures. Contributions to the global stiffness matrix from connections between boundary points from the K(bb) matrix are reported. The triangularization of a general K(bb) matrix on a parallel machine is specifically discussed.
Parametic investigations are performed using several two dimensional finite element formulations to determine their suitability for use in predicting extremum stresses in marine propellers. Comparisons are made of two NASTRAN elements (CTRIM6 and CTRAIA2) wherein elasticity properties have been modified to yield plane strain results. The accuracy of the elements is investigated by comparing finite element stress predictions with experimentally determined stresses in two classical cases: (1) tension in a flat plate with a circular hole; and (2) a filleted flat bar subjected to in-plane bending. The CTRIA2 element is found to provide good results. The displacement field from a three dimensional finite element model of a representative marine propeller is used as the boundary condition for the two dimensional plane strain investigations of stresses in the propeller blade and fillet. Stress predictions from the three dimensional analysis are compared with those from the two dimensional models. The validity of the plane strain modifications to the NASTRAN element is checked by comparing the modified CTRIA2 element stress predictions with those of the ABAQUS plane strain element, CPE4.
A study was completed of mathematically proper boundary conditions for unique numerical solution of internal, viscous, subsonic flows in the space shuttle main engine. The study has concentrated on well posed considerations, with emphasis on computational efficiency and numerically stable boundary condition statements. The method of implementing the established boundary conditions is applicable to a wide variety of finite difference and finite element codes, as demonstrated.
An approximation based on the representation of the boundary layer by lamina of uniform flow with suitable interlayer boundary conditions is shown to be accurate, efficient, and compatible with finite element formulations. The approximation has been implemented using existing codes to produce a model for assessing the suitability of the acoustic environment in a wind tunnel for the acoustic testing of propellers. It is found that, with suitable acoustic treatment and with measurements made near the propeller and well removed from the walls, the free field directivity and level can be reproduced with good fidelity.
A two-dimensional thermal model for spirally wound batteries has been developed. The governing equation of the model is the energy balance. Convective and insulated boundary conditions are used, and the equations are solved using a finite element code called TOPAZ2D. The finite element mesh is generated using a preprocessor to TOPAZ2D called MAZE. The model is used to estimate temperature profiles within a spirally wound D-size cell. The model is applied to the lithium/thionyl chloride cell because of the thermal management problems that this cell exhibits. Simplified one-dimensional models are presented that can be used to predict best and worst temperature profiles. The two-dimensional model is used to predict the regions of maximum temperature within the spirally wound cell. Normal discharge as well as thermal runaway conditions are investigated.
The mechanical properties of lunar regolith are significantly different from terrestrial soils; at low stress levels, the ultimate strength envelope is highly nonlinear, and the material is essentially dilatant. The effects of platen friction, membrane confinement, and material self weight in triaxial tests have been shown to be of great significance in low stress level tests. An inverse identification technique has been devised to account for these effects where the load-displacement response of the entire specimen, including the boundary conditions and the self-weight, was analyzed by nonlinear finite element techniques so as to obtain true material parameters.
This paper presents an aeroelastic trim drag optimization study of the Mach 0.745 Transonic Truss-Braced Wing (TTBW) aircraft with the Variable-Camber Continuous Trailing-Edge Flap (VCCTEF). An aero-structural analysis solver VSPAERO with transonic small disturbance, integral boundary-layer, and wing-strut interference corrections coupled to finite-element code BEAM3D is developed to provide a rapid aircraft aeroelastic performance evaluation. The VSPAERO model has been validated against wind tunnel test data. Aeroelastic trim drag optimization studies are conducted for a VCCTEF configuration with 6-spanwise sections. Three different flight conditions corresponding to Mach 0.745 are selected for the TTBW aircraft aeroelastic trim drag optimization at the design and off-design cruise lift coefficients. The preliminary optimization results show that the TTBW aircraft with the optimized VCCTEF deflection achieves a drag reduction of about 9.1 counts, 5.9 counts, and 11.8 counts corresponding to the lift coefficients 0.68, 0.73, and 0.78, respectively. A high-fidelity CFD solver FUN3D is used to validate the aeroelastic trim drag optimization.
Research into parachute performance continues to be a source of significant investment from the National Aeronautics and Space Administration to mitigate risks and to enable a variety of exploration missions, including landing on Mars as well as returning to Earth. The cost of flight tests to certify any changes to the current state-of-the-art parachute designs limits the development of next generation parachute systems. Fluid-structure interaction simulations could help accelerate this process once validated. The Launch, Ascent, and Vehicle Aerodynamics team is developing the capability to perform such fluid-structure interaction simulations by coupling a higher-order Cartesian immersed boundary computational fluid dynamics solver with adaptive mesh refinement to a finite element structural dynamics solver in space and time. We continue the effort to validate this tool with the Advanced Supersonic Parachute Inflation Research Experiments SR03 flight test featuring a strengthened parachute akin to the Mars 2020 mission that landed the Perseverance rover on Mars, and a higher freestream dynamic pressure prior to inflation. The effect of the flow conditions’ angle of attack and of the initial parachute shape are quantified. The impact of relaxing modeling assumptions with regards to radial stiffeners on the parachute canopy is also investigated. Results demonstrate improvements in agreement with the pull force recorded during the SR03 flight test as the initial conditions of the flow and parachute are brought closer to those experienced in flight, and further improved when the radial stiffener modeling assumptions are relaxed.
Research into parachute performance continues to be a source of significant investment from the National Aeronautics and Space Administration to mitigate risks and to enable a variety of exploration missions, including landing on Mars as well as returning to Earth. The cost of flight tests to certify any changes to the current state-of-the-art parachute designs limits the development of next generation parachute systems. Fluid-structure interaction simulations could help accelerate this process once validated. The Launch, Ascent, and Vehicle Aerodynamics team is developing the capability to perform such fluid-structure interaction simulations by coupling a higher-order Cartesian immersed boundary computational fluid dynamics solver with adaptive mesh refinement to a finite element structural dynamics solver in space and time. We continue the effort to validate this tool with the Advanced Supersonic Parachute Inflation Research Experiments SR03 flight test featuring a strengthened parachute akin to the Mars 2020 mission that landed the Perseverance rover on Mars, and a higher freestream dynamic pressure prior to inflation. The effect of the flow conditions’ angle of attack and of the initial parachute shape are quantified. The impact of relaxing modeling assumptions with regards to radial stiffeners on the parachute canopy is also investigated. Results demonstrate improvements in agreement with the pull force recorded during the SR03 flight test as the initial conditions of the flow and parachute are brought closer to those experienced in flight, and further improved when the radial stiffener modeling assumptions are relaxed.
New high-field accelerator magnets based on $Nb_3Sn$ superconductors are pushing the boundaries of magnet design and quench protection towards new limits. While their large stored energies and current densities result in a very challenging scenario for magnet protection, their great electromagnetic forces create also new requirements in terms of magnet design and stress management techniques. Furthermore, the strain sensitivity of $Nb_3Sn$ cables turns the electro-mechanical limits of the conductor into a parameter of the highest importance, where conductor degradation becomes a critical aspect in magnet operation. The coupling of all the above-mentioned considerations during quench is a case of special interest that adds further complexity to the design of $Nb_3Sn$ magnets. The objective of this paper is to provide a complete two-dimensional investigation of the coil and magnet structure mechanics during a quench event. Here, the analysis is performed using a combination of finite element codes that provide the necessary input for the mechanical study. The core of the modelling strategy relies on a thermal-electric model, whose results are directly used as loads in the mechanical simulation. In doing so, the stress evolution during and after a quench is obtained. We focus for this time in the analysis of a quench heater protected magnet, where dynamic effects are of less importance in contrast to other protection systems like CLIQ.
Selection of the best shape of a fillet in a tension bar such that no yielding occurs has long attracted the attention of engineers. Dimensions and notations for the bar and fillet are shown. With symmetry, only the upper half of the bar is considered. The optimal design problem is to find a boundary shape to minimize the total area of the fillet such that no yielding occurs. The classical boundary value problem is reduced to a variational or energy related problem which not only has excellent properties of existence and uniqueness but also provides the mathematical foundation for finite element analysis. The variational formulation may be viewed as the principle of virtual work and the finite element method as a application of the Galerkin method to the variational equation for approximate solution of the boundary value problem.
The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high- fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-re effective boundary conditions and wall modeling. The solver is based on high-order space-time - finite element methods. Continuous, discontinuous and C1-discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system.
The analysis of high frequency vibrations in plates is of particular interest in the study of structure borne noise in aircrafts. The current methods of analysis are either too expensive (finite element method) or may have a confidence band wider than desirable (Statistical Energy Analysis). An alternative technique to model the space and time averaged response of structural acoustics problems with enough detail to include all significant mechanisms of energy generation, transmission, and absorption is highly desirable. The focus of this paper is the development of a set of equations which govern the space and time averaged energy density in plates. To solve this equation, a new type of boundary value problem must be treated in terms of energy density variables using energy and intensity boundary conditions. A computer simulation verification study of the energy governing equation is performed. A finite element formulation of the new equations is also implemented and several test cases are analyzed and compared to analytical solutions.
Bolts and rivets provide a means of load transfer in the construction of aircraft. However, they give rise to stress concentrations and are often the source and location of static and fatigue failures. Furthermore, fastener holes are prone to cracks during take-off and landing. These cracks present the most common origin of structural failures in aircraft. Therefore, accurate determination of the contact stresses associated with such loaded holes in mechanically fastened joints is essential to reliable strength evaluation and failure prediction. As the laminate is subjected to loading, the contact region, whose extent is not known, develops between the fastener and the hole boundary through this contact region, which consists of slip and no-slip zones due to friction. The presence of the unknown contact stress distribution over the contact region between the pin and the composite laminate, material anisotropy, friction between the pin and the laminate, pin-hole clearance, combined bearing-bypass and shear loading, and finite geometry of the laminate result in a complex non-linear problem. In the case of bearing-bypass loading in compression, this non-linear problem is further complicated by the presence of dual contact regions. Previous research concerning the analysis of mechanical joints subjected to combined bearing-bypass and shear loading is non-existent. In the case of bearing-bypass loading only, except for the study conducted by Naik and Crews (1991), others employed the concept of superposition which is not valid for this non-linear problem. Naik and Crews applied a linear finite element analysis with conditions along the pin-hole contact region specified as displacement constraint equations. The major shortcoming of this method is that the variation of the contract region as a function of the applied load should be known a priori. Also, their analysis is limited to symmetric geometry and material systems, and frictionless boundary conditions. Since the contact stress distribution and the contact region are not known a priori, they did not directly impose the boundary conditions appropriate for modelling the contact and on-contact regions between the fastener and the hole. Furthermore, finite element analysis is not suitable for iterative design calculations for optimizing laminate construction in the presence of fasteners under complex loading conditions. In this study, the solution method developed by Madenci and Ileri (1992a,b) has been extended to determine the contact stresses in mechanical joints under combined bearing-bypass and shear loading, and bearing-bypass loading in compression resulting in dual contact regions.
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.
Small-scale mechanical testing has been used to isolate the properties of individual grain boundaries and regions of the lattice in undoped UO 2 polycrystals. Notched microcantilever specimens were prepared by focused ion beam milling and tested in bending. Fracture toughness values were calculated from finite element analysis of the sample geometry at fracture load using the J-integral method. Lattice and grain boundary fracture toughness values of 1.17 ± 0.16 MPa m 1/2 and 1.08 ± 0.08 MPa m 1/2 , respectively, were obtained. The similarity between the lattice and grain boundary fracture toughness values is consistent with the preference for transgranular room-temperature fracture in UO 2 . The grain boundary and lattice fracture toughness values also agree well with the toughness of dense polycrystals, which suggests that microstructural-scale toughening mechanisms are limited in such samples.
The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.