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NASA NTRS1991

Coupled fluid-structure interaction. Part 1: Theory. Part 2: Application

A general three dimensional variational principle is obtained for the motion of an acoustic field enclosed in a rigid or flexible container by the method of canonical decomposition applied to a modified form of the wave equation in the displacement potential. The general principle is specialized to a mixed two-field principle that contains the fluid displacement potential and pressure as independent fields. Semidiscrete finite element equations of motion based on this principle are derived and sample cases are given.

Felippa, Carlos A.↗

NASA NTRS1991

A geometric nonlinear degenerated shell element using a mixed formulation with independently assumed strain fields

A mixed formulation is chosen to overcome deficiencies of the standard displacement-based shell model. Element development is traced from the incremental variational principle on through to the final set of equilibrium equations. Particular attention is paid to developing specific guidelines for selecting the optimal set of strain parameters. A discussion of constraint index concepts and their predictive capability related to locking is included. Performance characteristics of the elements are assessed in a wide variety of linear and nonlinear plate/shell problems. Despite limiting the study to geometric nonlinear analysis, a substantial amount of additional insight concerning the finite element modeling of thin plate/shell structures is provided. For example, in nonlinear analysis, given the same mesh and load step size, mixed elements converge in fewer iterations than equivalent displacement-based models. It is also demonstrated that, in mixed formulations, lower order elements are preferred. Additionally, meshes used to obtain accurate linear solutions do not necessarily converge to the correct nonlinear solution. Finally, a new form of locking was identified associated with employing elements designed for biaxial bending in uniaxial bending applications.

Graf, Wiley E.↗

NASA NTRS1990

Efficient finite element modeling of laminated composite plates based on higher-order theory

A simple and efficient three-node plate bending element for the analysis of composite laminates is developed from a variational principle. The deformations due to transverse shear and transverse normal effects are accounted for, allowing accurate predictions in the range of thin to thick plates. The methodology incorporates C(exp 0) and C(exp 1) continuous displacement approximations and yields accurate ply-by-ply predictions of all displacement, strain, and stress variables.

Tessler, Alexander↗

NASA NTRS1991

A Lagrange multiplier based divide and conquer finite element algorithm

A novel domain decomposition method based on a hybrid variational principle is presented. Prior to any computation, a given finite element mesh is torn into a set of totally disconnected submeshes. First, an incomplete solution is computed in each subdomain. Next, the compatibility of the displacement field at the interface nodes is enforced via discrete, polynomial and/or piecewise polynomial Lagrange multipliers. In the static case, each floating subdomain induces a local singularity that is resolved very efficiently. The interface problem associated with this domain decomposition method is, in general, indefinite and of variable size. A dedicated conjugate projected gradient algorithm is developed for solving the latter problem when it is not feasible to explicitly assemble the interface operator. When implemented on local memory multiprocessors, the proposed methodology requires less interprocessor communication than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory and compares favorably with factorization based parallel direct methods.

Farhat, C.↗

NASA NTRS1992

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or 'drilling' degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not 'lock', possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.

Aminpour, M. A.↗

NASA NTRS1990

Localized basis functions and other computational improvements in variational nonorthogonal basis function methods for quantum mechanical scattering problems involving chemical reactions

The Generalized Newton Variational Principle for 3D quantum mechanical reactive scattering is briefly reviewed. Then three techniques are described which improve the efficiency of the computations. First, the fact that the Hamiltonian is Hermitian is used to reduce the number of integrals computed, and then the properties of localized basis functions are exploited in order to eliminate redundant work in the integral evaluation. A new type of localized basis function with desirable properties is suggested. It is shown how partitioned matrices can be used with localized basis functions to reduce the amount of work required to handle the complex boundary conditions. The new techniques do not introduce any approximations into the calculations, so they may be used to obtain converged solutions of the Schroedinger equation.

Schwenke, David W.↗

NASA NTRS1992

r-adaptive methods based on element-level error indicators for parallel analysis of plates and shells

The formulation and application of element-level error indicators based on parametrized variational principles are investigated. These indicators are ideally suited to drive adaptive mesh refinment on parallel computers where access to neighboring elements resident on different processors may incur significant computational overhead. Furthermore, such estimators are not affected by physical jumps at wavefronts or interfaces. An estimator derived from the higher-order element energy is applied to r adaptation of meshes in plates and shell structures where the exact solution is known.

Militello, Carmelo↗

NASA NTRS1992

Stress analysis of a rectangular implant in laminated composites using 2-D and 3-D finite elements

An analysis method using the FEM based on the Hellinger-Reissner variation principle has been developed to determine the 3-D stresses and displacements near a rectangular implant inside a laminated composite material. Three-dimensional elements are employed in regions where the interlaminar stress is considered to be significant; 2-D elements are used in other areas. Uniaxially loaded graphite-epoxy laminates have been analyzed; the implant was modeled as four plies of 3501/6 epoxy located in the middle of the laminate. It is shown that the interlaminar stresses are an order of magnitude lower than the stress representing the applied far-field load. The stress concentration factors of both the interlaminar and in-plane stresses depend on the stacking sequence of the laminate.

Chow, Wai T.↗

NASA NTRS1992

Dynamic characterization, monitoring and control of rotating flexible beam-mass structures via piezo-embedded techniques

A variational principle and a finite element discretization technique were used to derive the dynamic equations for a high speed rotating flexible beam-mass system embedded with piezo-electric materials. The dynamic equation thus obtained allows the development of finite element models which accommodate both the original structural element and the piezoelectric element. The solutions of finite element models provide system dynamics needed to design a sensing system. The characterization of gyroscopic effect and damping capacity of smart rotating devices are addressed. Several simulation examples are presented to validate the analytical solution.

Lai, Steven H.-Y.↗

NASA NTRS1993

Probabilistic finite elements for fatigue and fracture analysis

An overview of the probabilistic finite element method (PFEM) developed by the authors and their colleagues in recent years is presented. The primary focus is placed on the development of PFEM for both structural mechanics problems and fracture mechanics problems. The perturbation techniques are used as major tools for the analytical derivation. The following topics are covered: (1) representation and discretization of random fields; (2) development of PFEM for the general linear transient problem and nonlinear elasticity using Hu-Washizu variational principle; (3) computational aspects; (4) discussions of the application of PFEM to the reliability analysis of both brittle fracture and fatigue; and (5) a stochastic computational tool based on stochastic boundary element (SBEM). Results are obtained for the reliability index and corresponding probability of failure for: (1) fatigue crack growth; (2) defect geometry; (3) fatigue parameters; and (4) applied loads. These results show that initial defect is a critical parameter.

Belytschko, Ted↗

NASA NTRS1992

Membrane triangles with corner drilling freedoms. I - The EFF element

The formulation of 3-node 9-DOF membrane elements with normal-to-element-plane rotations (drilling freedoms) is examined in the context of parametrized variational principles. In particular, attention is given to the application of the extended free formulation (EFF) to the construction of a triangular membrane element with drilling freedoms that initially has complete quadratic polynomial expansions in each displacement component. The main advantage of the EFF over the free formulation triangle is that an explicit form is obtained for the higher-order stiffness.

Alvin, Ken↗

NASA NTRS1992

Membrane triangles with corner drilling freedoms. II - The ANDES element

This is the second article in a three-part series on the construction of 3-node, 9-dof membrane elements with normal-to-its-plane rotational freedoms (the so-called drilling freedoms) using parametrized variational principles. In this part, one such element is derived within the context of the assumed natural deviatoric strain (ANDES) formulation. The higher-order strains are obtained by constructing three parallel-to-sides pure-bending modes from which natural strains are obtained at the corner points and interpolated over the element. To attain rank sufficiency, an additional higher-order 'torsional' mode, corresponding to equal hierarchical rotations at each corner with all other motions precluded, is incorporated. The resulting formulation has five free parameters. When these parameters are optimized against pure bending by energy balance methods, the resulting element is found to coalesce with the optimal EFF element derived in Part I. Numerical integration as a strain filtering device is found to play a key role in this achievement.

Felippa, Carlos A.↗

NASA NTRS1992

Membrane triangles with corner drilling freedoms. III - Implementation and performance evaluation

This paper completes a three-part series on the formulation of 3-node, 9-dof membrane triangles with corner drilling freedoms based on parametrized variational principles. The first four sections cover element implementation details including determination of optimal parameters and treatment of distributed loads. Then three elements of this type, labeled ALL, FF and EFF-ANDES, are tested on standard plane stress problems. ALL represents numerically integrated versions of Allman's 1988 triangle; FF is based on the free formulation triangle presented by Bergan and Felippa in 1985; and EFF-ANDES represent two different formulations of the optimal triangle derived in Parts I and II. The numerical studies indicate that the ALL, FF and EFF-ANDES elements are comparable in accuracy for elements of unitary aspect ratios. The ALL elements are found to stiffen rapidly in inplane bending for high aspect ratios, whereas the FF and EFF elements maintain accuracy. The EFF and ANDES implementations have a moderate edge in formation speed over the FF.

Felippa, Carlos A.↗

NASA NTRS1993

Ab initio method for calculating total cross sections

A method for calculating total cross sections without formally including nonelastic channels is presented. The idea is to use a one channel T-matrix variational principle with a complex correlation function. The derived T matrix is therefore not unitary. Elastic scattering is calculated from T-parallel-squared, but total scattering is derived from the imaginary part of T using the optical theorem. The method is applied to the spherically symmetric model of electron-hydrogen scattering. No spurious structure arises; results for sigma(el) and sigma(total) are in excellent agreement with calculations of Callaway and Oza (1984). The method has wide potential applicability.

Bhatia, A. K.↗

NASA NTRS1993

A mixed formulation for interlaminar stresses in dropped-ply laminates

A structural model is developed for the linear elastic response of structures consisting of multiple layers of varying thickness such as laminated composites containing internal ply drop-offs. The assumption of generalized plane deformation is used to reduce the solution domain to two dimensions while still allowing some out-of-plane deformation. The Hellinger-Reissner variational principle is applied to a layerwise assumed stress distribution with the resulting governing equations solved using finite differences.

Harrison, Peter N.↗

NASA NTRS1993

Implementation of a mesh adaptive scheme based on an element-level error indicator

We investigate the formulation and application of element-level error indicators based on parametrized variational principles. The qualifier 'element-level' means that no information from adjacent elements is used for error estimation. This property is ideally suited to drive adaptive mesh refinement on parallel computers where access to neighboring elements resident on different processors may incur significant computational overhead. Furthermore, such indicators are not affected by physical jumps at junctures or interfaces. An element-level indicator has been derived from the higher-order element energy and applied to r and h mesh adaptation of meshes in plates and shell structures. We report on our initial experiments with a cylindrical shell that intersects with fist plates forming a simplified 'wing-body intersection' benchmark problem.

Keating, Scott↗

NASA NTRS1993

State-constrained booster trajectory solutions via finite elements and shooting

This paper presents an extension of a FEM formulation based on variational principles. A general formulation for handling internal boundary conditions and discontinuities in the state equations is presented, and the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions which only touch the state constraint and solutions which have a boundary arc of finite length are considered. Suitable shape and test functions are chosen for a FEM discretization. All element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form. The final form of the algebraic equations is then derived. A simple state-constrained problem is solved. Then, for a practical application of the use of the FEM formulation, a launch vehicle subject to a dynamic pressure constraint (a first-order state inequality constraint) is solved. The results presented for the launch-vehicle trajectory have some interesting features, including a touch-point solution.

Bless, Robert R.↗

NASA NTRS1993

Transverse cracking and stiffness reduction in composite laminates

A study of transverse cracking mechanism in composite laminates is presented using a singular hybrid finite element model. The model provides the global structural response as well as the precise local crack-tip stress fields. An elasticity basis for the problem is established by employing Lekhnitskii's complex variable potentials and method of eigenfunction expansion. Stress singularities associated with the transverse crack are obtained by decomposing the deformation into the symmetric and antisymmetric modes and proper boundary conditions. A singular hybrid element is thereby formulated based on the variational principle of a modified hybrid functional to incorporate local crack singularities. Axial stiffness reduction due to transverse cracking is studied. The results are shown to be in very good agreement with the existing experimental data. Comparison with simple shear lag analysis is also given. The effects of stress intensity factors and strain energy density on the increase of crack density are analyzed. The results reveal that the parameters approach definite limits when crack densities are saturated, an evidence of the existence of characteristic damage state.

Yuan, F. G.↗