On the stability of systems with monotone and odd monotone nonlinearities.
Stability of feedback systems with monotone and odd monotone nonlinearities
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On a numerical sufficiency test for monotonic convergence of finite element models
Finite element analyses characterized by monotonic convergence include the discipline for meaningful measurements of convergence rate and consequently economical extrapolation. Few proposers of element models guarantee monotonic convergence for their elements. Thus, a need exists for an automatic test to classify available element models. This paper describes such a test - a test can be performed using a digital computer to guarantee that a particular element model imbues monotonicity. It describes the test and its basis. It examines seven element models for a rectangular membrane to illustrate the value of the tests. Besides confirming results already known, the application yields new data. It 'proves' monotonicity for two improved models, defines the range of element proportions for which another element can be guaranteed to exhibit monotonicity, and suggests that another element is deficient. In the special case of absolutely convergent membrane displacement models, proof of monotonicity is a necessary and sufficient condition to insure that upper bound estimates of strain energy are developed. Accordingly, the test furnishes a proof of bound solutions independently of requirements on displacement continuity the element basis may or may not satisfy.
Melosh, R. J.↗
Non-Monotonic Response of the Climate System to Abrupt CO2 Forcing
We explore the climate system response to abrupt CO2 forcing, spanning the range 1 × to 8×CO2, with two state-of-the-art coupled atmosphere-ocean-sea-ice-land models: the NASA Goddard Institute for Space Studies Model E2.1-G (GISS-E2.1-G) and the Community Earth System Model (CESM-LE). We find that the effective climate sensitivity is a non-monotonic function of CO2 in both models, reaching a minimum at 3×CO2 for GISS-E2.1-G, and 4×CO2 for CESM-LE. A similar non-monotonic response is found in Northern Hemisphere surface temperature, sea-ice, precipitation, the latitude of zero precipitation minus-evaporation, and the strength of the Hadley cell. Interestingly, the Atlantic meridional overturning circulation collapses when non-monotonicity appears and does not recover for larger CO2 forcings. Analyzing the climate response over the same CO2 range with slab ocean versions of the same models, we demonstrate that the climate system's non-monotonic response is linked to ocean dynamics.
Non-monotonic response↗
Accurate monotone cubic interpolation
Monotone piecewise cubic interpolants are simple and effective. They are generally third-order accurate, except near strict local extrema where accuracy degenerates to second-order due to the monotonicity constraint. Algorithms for piecewise cubic interpolants, which preserve monotonicity as well as uniform third and fourth-order accuracy are presented. The gain of accuracy is obtained by relaxing the monotonicity constraint in a geometric framework in which the median function plays a crucial role.
Huynh, Hung T.↗
Monotone implicit algorithms for the small-disturbance and full potential equations applied to transonic flows
Numerical calculations of transonic flows by potential equations typically use algorithms that change the method of calculation for regions of subsonic and supersonic flow. In this paper, implicit approximate-factorization algorithms are modified to use the monotonic switch in the type of finite-differencing that was developed by Godunov for the Euler equations. Calculations of flows over airfoils by these algorithms are compared with calculations by other methods that are in common usage. For the small-disturbance potential equation, comparisons are made with the Murman-Cole method and the monotone method of Engquist and Osher for both steady and unsteady flows. For the full potential equation, comparisons are made with the methods of Jameson and of Holst and Ballhaus for steady flows. The comparisons show that the monotone methods are more stable. For steady flows, solutions are obtained for cases where the Murman-Cole switch requires a time step over ten times smaller in order for the calculations to remain stable. These improvements are achieved with no increase in computer storage and only minor modifications in current codes.
Goorjian, P. M.↗
Accurate upwind-monotone (nonoscillatory) methods for conservation laws
The well known MUSCL scheme of Van Leer is constructed using a piecewise linear approximation. The MUSCL scheme is second order accurate at the smooth part of the solution except at extrema where the accuracy degenerates to first order due to the monotonicity constraint. To construct accurate schemes which are free from oscillations, the author introduces the concept of upwind monotonicity. Several classes of schemes, which are upwind monotone and of uniform second or third order accuracy are then presented. Results for advection with constant speed are shown. It is also shown that the new scheme compares favorably with state of the art methods.
Huynh, Hung T.↗
Matrix cracking in laminated composites under monotonic and cyclic loadings
An analytical model based on the internal state variable (ISV) concept and the strain energy method is proposed for characterizing the monotonic and cyclic response of laminated composites containing matrix cracks. A modified constitution is formulated for angle-ply laminates under general in-plane mechanical loading and constant temperature change. A monotonic matrix cracking criterion is developed for predicting the crack density in cross-ply laminates as a function of the applied laminate axial stress. An initial formulation for a cyclic matrix cracking criterion for cross-ply laminates is also discussed. For the monotonic loading case, a number of experimental data and well-known models are compared with the present study for validating the practical applicability of the ISV approach.
Allen, David H.↗
Comparison of Mode II and III Monotonic and Fatigue Delamination Onset Behavior for Carbon/Toughened Epoxy Composites
Monotonic and fatigue tests were performed to compare the Mode II and III interlaminar fracture toughness and fatigue delamination onset for Tenax-HTA/R6376 carbon/toughened epoxy composites. The Mode II interlaminar fracture toughness and fatigue delamination onset were characterized using the end-notched flexure (ENF) test while the Mode III interlaminar fracture toughness and fatigue delamination onset were characterized by using the edge crack torsion (ECT) test. Monotonic tests show that the Mode III fracture toughness is higher than the Mode II fracture toughness. Both Mode II and III cyclic loading greatly increases the tendency for a delamination to grow relative to a single monotonically increasing load. Under fatigue loading, the Mode III specimen also has a longer life than the Mode II specimen.
Li, Jian↗
Applications of partial orderings to the study of positive definiteness, monotonicity, and convergence of iterative methods for linear systems.
Most of the known results concerning convergence of iterative methods for solving linear systems involve either positive definiteness or monotonicity. In this paper a new concept, called K-semipositivity, is introduced, which provides a link between convergence theory, monotonicity, and positive definiteness. By using this concept, together with partial orderings in Euclidean n-space, several new convergence theorems are proved. Application to Jacobi's methods and the theory of regular splittings shows the usefulness of these new results.
Vandergraft, J. S.↗
Life prediction of materials exposed to monotonic and cyclic loading: Bibliography
This bibliography is comprised of approximately 1200 reference citations related to the mechanics of failure in aerospace structures. Most of the references are for information on life prediction for materials exposed to monotonic and cyclic loading in elevated temperature environments such as that in the hot end of a gas turbine engine. Additional citations listed are for documents on the thermal and mechanical effects on solar cells in the cryogenic vacuum environment; radiation effects on high temperature mechanical properties; and high cycle fatigue technology as applicable to gas turbine engine bearings. The bibliography represents a search of the literature published in the period April 1962 through April 1974 and is largely limited to documents published in the United States. It is a companion volume to NASA CR-134750, Life Prediction of Materials Exposed to Monotonic and cyclic Loading - A Technology Survey.
Carpenter, J. L., Jr.↗
Finite-element analysis of crack growth under monotonic and cyclic loading
An elastic-plastic (incremental) finite-element analysis, in conjunction with a crack-growth criterion, was used to study crack-growth behavior under monotonic and cyclic loading. The crack-growth criterion was based on crack-tip strain. Whenever the crack-tip strain equals or exceeds a critical strain value, the crack grows. The effects of element-mesh size, critical strain, strain hardening, and specimen type (tension or bending) on crack growth under monotonic loading were investigated. Crack growth under cyclic loading (constant amplitude and simple variable amplitude) were also studied. A combined hardening theory, which incorporates features of both isotropic and kinematic hardening under cyclic loading, was also developed for smooth yield surfaces and was used in the analysis.
Newman, J. C., Jr.↗
Implicit calculations of transonic flows using monotone methods
Implicit approximate-factorization algorithms have been developed that use monotone methods for the calculation of steady and unsteady transonic flows governed by the small-disturbance-potential equation. These algorithms use the new Engquist-Osher switch in the type-dependent differencing in place of the standard Murman-Cole switch. The resulting algorithms are more stable; hence, calculations can be done more efficiently. For steady flows, the convergence rate is about 35% faster, and for unsteady flows the allowable time step is about 10 times larger. These improvements are achieved with no increase in computer storage and with only minor modifications in codes that use the Murman-Cole switch. Also an implicit algorithm has been developed for the steady full-potential equation in one-dimension, which uses monotone methods.
Goorjian, P. M.↗
A review of failure models for unidirectional ceramic matrix composites under monotonic loads
Ceramic matrix composites offer significant potential for improving the performance of turbine engines. In order to achieve their potential, however, improvements in design methodology are needed. In the past most components using structural ceramic matrix composites were designed by trial and error since the emphasis of feasibility demonstration minimized the development of mathematical models. To understand the key parameters controlling response and the mechanics of failure, the development of structural failure models is required. A review of short term failure models with potential for ceramic matrix composite laminates under monotonic loads is presented. Phenomenological, semi-empirical, shear-lag, fracture mechanics, damage mechanics, and statistical models for the fast fracture analysis of continuous fiber unidirectional ceramic matrix composites under monotonic loads are surveyed.
Tripp, David E.↗
A review of failure models for ceramic matrix composite laminates under monotonic loads
Ceramic matrix composites offer significant potential for improving the performance of turbine engines. In order to achieve their potential, however, improvements in design methodology are needed. In the past most components using structural ceramic matrix composites were designed by trial and error since the emphasis of feasibility demonstration minimized the development of mathematical models. To understand the key parameters controlling response and the mechanics of failure, the development of structural failure models is required. A review of short term failure models with potential for ceramic matrix composite laminates under monotonic loads is presented. Phenomenological, semi-empirical, shear-lag, fracture mechanics, damage mechanics, and statistical models for the fast fracture analysis of continuous fiber unidirectional ceramic matrix composites under monotonic loads are surveyed.
Tripp, David E.↗
Effect of a multi-dimensional flux function on the monotonicity of Euler and Navier-Stokes computations
A limiting method has been devised for a grid-independent flux function for use with the two-dimensional Euler and Navier-Stokes equations. This limiting is derived from a monotonicity analysis of the model and allows for solutions with reduced oscillatory behavior while still maintaining sharper resolution than a grid-aligned method. In addition to capturing oblique waves sharply, the grid-independent flux function also reduces the entropy generated over an airfoil in an Euler computation and reduces pressure distortions in the separated boundary layer of a viscous-flow airfoil computation. The model has also been extended to three dimensions, although no angle-limiting procedure for improving monotonicity characteristics has been incorporated.
Rumsey, Christopher L.↗
A nearly-monotone genuinely multidimensional scheme for the Euler equations
Recent progress made in the development of a genuinely multidimensional finite-volume method for the Euler equations is reported. In this method, wave information is derived from a reconstruction procedure applied to the data on a triangle. The algorithm is therefore best suited to application on an unstructured grid. A flux formula which provides nearly-monotone solutions is also presented. The new method is tested on three simple 2D flows, and the results show that dominant-wave resolution is high, and that strong-wave transitions are nearly monotone.
Parpia, Ijaz H.↗
Reformulating Non-Monotonic Theories for Inference and Updating
We aim to help build programs that do large-scale, expressive non-monotonic reasoning (NMR): especially, 'learning agents' that store, and revise, a body of conclusions while continually acquiring new, possibly defeasible, premise beliefs. Currently available procedures for forward inference and belief revision are exhaustive, and thus impractical: they compute the entire non-monotonic theory, then re-compute from scratch upon updating with new axioms. These methods are thus badly intractable. In most theories of interest, even backward reasoning is combinatoric (at least NP-hard). Here, we give theoretical results for prioritized circumscription that show how to reformulate default theories so as to make forward inference be selective, as well as concurrent; and to restrict belief revision to a part of the theory. We elaborate a detailed divide-and-conquer strategy. We develop concepts of structure in NM theories, by showing how to reformulate them in a particular fashion: to be conjunctively decomposed into a collection of smaller 'part' theories. We identify two well-behaved special cases that are easily recognized in terms of syntactic properties: disjoint appearances of predicates, and disjoint appearances of individuals (terms). As part of this, we also definitionally reformulate the global axioms, one by one, in addition to applying decomposition. We identify a broad class of prioritized default theories, generalizing default inheritance, for which our results especially bear fruit. For this asocially monadic class, decomposition permits reasoning to be localized to individuals (ground terms), and reduced to propositional. Our reformulation methods are implementable in polynomial time, and apply to several other NM formalisms beyond circumscription.
Grosof, Benjamin N.↗
An Object-oriented Query Processor that Produces Monotonically Improving Approximate Answers
The paper describes an object-oriented query processor that makes approximate answers available if there is not enough time to produce an exact answer or if part of the database is unavailable. The accuracy of the approximate result produces improves monotonically with the amount of data retrieved to produce the result. The query processing algorithm is based on an approximate relational data model and works within a standard relational algebra framework. The query processor maintains an object-oriented view on an underlying level and can be implemented on a relational database system with little change to the relational architecture. We show how a monotone query processing strategy can be implemented, making effective use of semantic information presented by the object-oriented view.
Vrbsky, S. V.↗