Liping Liu

Eighth-order methods with high efficiency index for solving nonlinear equations

In this paper, we derive a new family of eighth-order methods for solving simple roots of nonlinear equations by using weight function methods. Per iteration these methods require three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency indexes are 1.682. Numerical comparisons are made to show the performance of the derived methods, as shown in the illustration examples.

Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface

The nonlinear aeroelastic response of a two-dimensional airfoil, including a control surface with freeplay placed in an incompressible flow, is studied. The model equations are formulated as a set of first-order ordinary differential equations. First, the dynamic response is investigated by a time integration method, and the time integration results are used for the verification of the harmonic balance results. The interesting hysteresis phenomenon and the effect of initial conditions of the subcritical bifurcation are presented. A higher-order harmonic balance method is then derived to investigate the high harmonics of the airfoil motions. The harmonic balance prediction is verified by comparison to the results from a numerical time marching integration and also by comparison to results from a previous experiment. Nomenclature a = nondimensional distance of the elastic axis from the midchord, with respect to the semichord B = damping submatrix b = semichord Ch = stiffness (per unit span) of wing in deflection C(k) = generalized Theodorsen function Cα, Cβ = torsional stiffness (per unit span) of wing around a and of aileron around c c = nondimensional distance of the control surface (aileron) hinge line from the midchord, with respect to the semichord

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traubs conjecture [7] for four function evaluations per iteration. Notice that Bi et al.s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newtons method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kous method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816-1821] and Wangs method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14-19] are the special cases of the new improvements.

Higher Order Harmonic Balance Analysis for Limit Cycle Oscillations in an Airfoil with Cubic Restoring Forces

This is a study of a two dimensional airfoil including a cubic spring stifiness placed in an incompressible ∞ow. A new formulation of the harmonic balance method is employed for the aeroelastic airfoil to investigate the amplitude and frequency of the limit cycle oscillations. The results are compared with the results from the classical harmonic balance approach and from the conventional time marching integration method.

Optimizing genomic medicine in epilepsy through a gene-customized approach to missense variant interpretation

Gene panel and exome sequencing have revealed a high rate of molecular diagnoses among diseases where the genetic architecture has proven suitable for sequencing approaches, with a large number of distinct and highly penetrant causal variants identified among a growing list of disease genes. The challenge is, given the DNA sequence of a new patient, to distinguish disease-causing from benign variants. Large samples of human standing variation data highlight regional variation in the tolerance to missense variation within the protein-coding sequence of genes. This information is not well captured by existing bioinformatic tools, but is effective in improving variant interpretation. To address this limitation in existing tools, we introduce the missense tolerance ratio (MTR), which summarizes available human standing variation data within genes to encapsulate population level genetic variation. We find that patient-ascertained pathogenic variants preferentially cluster in low MTR regions (P < 0.005) of well-informed genes. By evaluating 20 publicly available predictive tools across genes linked to epilepsy, we also highlight the importance of understanding the empirical null distribution of existing prediction tools, as these vary across genes. Subsequently integrating the MTR with the empirically selected bioinformatic tools in a gene-specific approach demonstrates a clear improvement in the ability to predict pathogenic missense variants from background missense variation in disease genes. Among an independent test sample of case and control missense variants, case variants (0.83 median score) consistently achieve higher pathogenicity prediction probabilities than control variants (0.02 median score; Mann-Whitney U test, P < 1 × 10-16). We focus on the application to epilepsy genes; however, the framework is applicable to disease genes beyond epilepsy.

Origin of the pre-tropical storm Debby (2006) African easterly wave-mesoscale convective system

The origins of the pre-Debby (2006) mesoscale convective system (MCS) and African easterly wave (AEW) and their precursors were traced back to the southwest Arabian Peninsula, Asir Mountains (AS), and Ethiopian Highlands (EH) in the vicinity of the ITCZ using satellite imagery, GFS analysis data and ARW model. The sources of the convective cloud clusters and vorticity perturbations were attributed to the cyclonic convergence of northeasterly Shamal wind and the Somali jet, especially when the Mediterranean High shifted toward east and the Indian Ocean high strengthened and its associated Somali jet penetrated farther to the north. The cyclonic vorticity perturbations were strengthened by the vorticity stretching associated with convective cloud clusters in the genesis region—southwest Arabian Peninsula. A conceptual model was proposed to explain the genesis of convective cloud clusters and cyclonic vorticity perturbations preceding the pre-Debby (2006) AEW–MCS system.

Orographic Influence on Basic Flow and Cyclone Circulation and Their Impacts on Track Deflection of an Idealized Tropical Cyclone

AbstractA series of idealized numerical experiments and vorticity budget analyses is performed to examine several mechanisms proposed in previous studies to help understand the orographic influence on track deflection over a mesoscale mountain range. When an idealized tropical cyclone (TC) is embedded in a uniform, easterly flow and passes over a mountain with a moderate Froude number, it is deflected to the south upstream, moves over the mountain anticyclonically, and then resumes its westward movement. The vorticity budget analysis indicates that the TC movement can be predicted by the maximum vorticity tendency (VT). The orographic effects on the above TC track deflection are explained by the following: 1) Upstream of the mountain, the easterly basic flow is decelerated as a result of orographic blocking that causes the flow to become subgeostrophic, which advects the TC to the southwest, analogous to the advection of a point vortex embedded in a flow. The VT is primarily dominated by the horizontal vo...

Effects of Landfall Location and Approach Angle of an Idealized Tropical Cyclone over a Long Mountain Range

Effects of landfall location and approach angle on track deflection associated with a tropical cyclone (TC) passing over an idealized and Central Appalachian Mountain is investigated by a series of idealized numerical experiments. When the TC landfalls on the central portion of the mountain range, it is deflected to the south upstream, passes over the mountain anticyclonically, and then moves westward downstream. The TC motion is steered by the positive vorticity tendency (VT) which is dominated by horizontal vorticity advection upstream and downstream, but with additional influence from the stretching and residual terms, which are mainly associated with diabatic heating and frictional effects. The track deflection mechanism upstream and downstream is similar to the dry flow in previous study, but is very different in the vicinity of the mountain. When the TC landfalls near the northern (southern) tip, it experiences less (more) southward deflection due to stronger (weaker) vorticity advection around the tip. When the TC approaches the mountain range from the southeast and landfalls on the northern tip, center, or southern tip, the track deflections are similar to those embedded in an easterly flow but with weaker orographic blocking. These results are similar to the cases simulated in the dry flow in previous study, except that there is no track discontinuity due to the weaker orographic blocking associated with strong TC convection. When a TC moves along the north-south mountain range from the south, it tends to deflect toward the mountain and then crosses over to the other side at later time. In these cases, the positive VT is influenced by all horizontal vorticity advection, vorticity stretching (diabatic heating) and residual (friction) terms due to longer and stronger interaction with the mountain range. The vorticity stretching is mainly caused by diabatic heating in the moist flow, instead of by lee slope vorticity stretching in the previous study for dry flow.