Analytical Weight Function Research Articles

Continuous approximation for linear combination coefficients: Exploring a neglected concept

An alternative approach is introduced for accurately describe atomic orbitals using an integral transform of Gaussian-type basis functions (GTO) combined with continuous and analytical weight functions. The functional form of these weight functions was optimized using the variational criterion, achieving μHartree-level precision in energy calculations for atoms, monovalent ions up to the third period of the periodic table, and the H2 molecule. However, errors increased for heavier atoms, such as sulfur and chlorine, indicating the need for further refinement of the weight functions. This approach enhances the accuracy of electronic structure calculations, offering a rigorous alternative for deriving one-electron wave functions from discrete basis sets and vice-versa for atoms and molecules.

Weight function analyses and stress intensity factors for corner/surface crack in straight and tapered lugs

AbstractLug is one of the most important lap joints in many engineering structures, especially airframes. Through‐thickness and part‐through cracks emanating from pin‐loaded lug hole are frequently encountered due to stress concentration. Stress intensity factors for various crack geometries are the prerequisite for damage tolerance design of attachment lug. In this paper, analytical 2D weight functions for through‐thickness crack in lug are derived considering the contact condition between the pin and lug. With the derived 2D weight function, a highly efficient 3D slice synthesis weight function method (SSWFM) is then developed to determine 3D SIFs of hole‐edge corner/surface crack in straight and tapered lugs under various pin loadings. The resulting 3D SIFs are verified by comparisons with the numerical FEM/Franc3D solutions and literature data. The SSWFM is about 1800 times faster than the numerical FEM/Franc3D and therefore will significantly enhance damage tolerance analysis capability for attachment lugs.

Determination of complex stress intensity factors for interface cracks in bi-material specimens subjected to ununiform stresses

This paper presents the calculation of complex stress intensity factors (K) for interface cracks in commonly used bi-material specimens subjected to arbitrary ununiform stress fields. First, detailed finite element analyses were carried out for cracks in centre-cracked plate (CCP) and single edge-cracked plate (SECP) bi-material specimens of different material combinations under non-uniform stress distributions. The complex SIFs were determined for the loading cases of uniform, linear, parabolic or cubic stress distributions applied to the crack faces in both normal and shear directions. Then, general mathematical forms of the two complex crack face weight functions for interface cracks, one for normal loading, and one for shear loading, have been formulated by taking advantages of the knowledge of specific analytical complex weight functions available in the literature, and certain properties of the complex weight functions in general. The existence of the general functional forms simplifies the determination of these weight functions for specific crack geometrical configurations and bi-material combinations. The determination of a specific weight function is thus reduced to the determination of the unknown parameters of the generalized weight function functional forms. It was demonstrated these unknown parameters can be determined from reference stress intensity factors. This method was used to derive the complex weight functions for interface cracks in bi-material CCP and SECP specimens. The derived closed-form weight functions were then validated against complex stress intensity factors solutions for several non-linear normal and shear stress distributions.

SIF and CMOD for three-point bending beams with arbitrary span-to-width ratios by using analytical weight function method

SIF and CMOD for three-point bending beams with arbitrary span-to-width ratios by using analytical weight function method

A review and verification of analytical weight function methods in fracture mechanics

AbstractThe weight function method is a powerful method for the determination of key parameters of stress intensity factors and crack opening displacements that are fundamental for crack analyses in fracture mechanics. The method has a remarkable computational efficiency without compromising solution accuracy and is especially suited for cracks in real‐world components with complex stress gradients. A critical issue for successful development and reliable application of the weight function method is how to achieve verifiable and high solution accuracy. This paper provides several reflections on the current state of the art in weight function methods, including a brief historical perspective and description of several analytical and numerical weight methods, with main focus on the two‐dimensional (2D) analytical approaches. A variety of typical application examples are presented. A platform for point‐by‐point accuracy assessment by using the Green's functions is established on a completely independent numerical approach with validated accuracy. Rigorous and systematic verification of the accuracy levels of different 2D analytical weight function approaches are conducted using several benchmark crack geometries. The sources of inaccuracy of different analytical weight function methods are analysed. A brief account on weight function methods for analysing three‐dimensional (3D) crack problems with arbitrary univariant and bivariant stress distributions is provided.

Review and evaluation of weight functions and stress intensity factors for edge-cracked finite-width plate

Various analytical and numerical weight functions (WFs) for the edge-cracked finite-width plate (ECFWP) configuration have been developed in the literature for efficient fracture mechanics analyses involving complicated/arbitrary load conditions. The objective of this paper is to make a comprehensive review and critical evaluation of the existing WFs determined by using various approaches, e.g. singular integral equation methods; one/two reference load case(s) based analytical approaches; fitting and interpolation to handbook solutions; numerical methods such as the complex Taylor series expansion and finite element analysis. The accuracy levels of existing WFs are assessed in rigorous manner by benchmarking the corresponding Green’s functions (GFs) against well-recognized accurate solutions. A highly accurate and wide-ranging analytical ECFWP WF is presented in closed form. Stress intensity factor and crack opening displacement solutions for various load cases of practical interest are easily determined with high accuracy by using the well-verified analytical ECFWP WF for this crack configuration to demonstrate the methods.

Evaluation of various analytical weight function methods base on exact K-solutions of an edge-cracked circular disc

Weight function method is a powerful tool for the determination of key parameters such as stress intensity factors and crack opening displacements for cracks in complicated stress fields. Accurate determination of the weight functions is of primary importance for successful applications of the method. This paper makes detailed accuracy evaluation of two kinds of widely used approximate (2D) weight function approaches, one being based on crack opening displacement for one reference stress, the other on two reference stresses coupled with a geometric condition. To eliminate possible sources of error, the edge-cracked circular disc with exactK-solutions is used for deriving the two kinds of weight functions and for benchmarking against the two weight function approaches. A highly accurate numerical weight function method using the complex Taylor series expansion and complex finite element computation is also used for assessing the related Green’s functions. Results from the evaluation will be instructive for accurate determination of weight functions for other finite edge-crack geometries.

Two-dimensional analytic weighting functions for limb scattering

Through the inversion of limb scatter measurements it is possible to obtain vertical profiles of trace species in the atmosphere. Many of these inversion methods require what is often referred to as weighting functions, or derivatives of the radiance with respect to concentrations of trace species in the atmosphere. Several radiative transfer models have implemented analytic methods to calculate weighting functions, alleviating the computational burden of traditional numerical perturbation methods. Here we describe the implementation of analytic two-dimensional weighting functions, where derivatives are calculated relative to atmospheric constituents in a two-dimensional grid of altitude and angle along the line of sight direction, in the SASKTRAN-HR radiative transfer model. Two-dimensional weighting functions are required for two-dimensional inversions of limb scatter measurements. Examples are presented where the analytic two-dimensional weighting functions are calculated with an underlying one-dimensional atmosphere. It is shown that the analytic weighting functions are more accurate than ones calculated with a single scatter approximation, and are orders of magnitude faster than a typical perturbation method. Evidence is presented that weighting functions for stratospheric aerosols calculated under a single scatter approximation may not be suitable for use in retrieval algorithms under solar backscatter conditions.

Shot Peening Effect on Fatigue Crack Repaired Weldments

Fracture mechanics calculations are required to validate the safety level defined in design codes to prevent a fatigue failure. The periodic inspection-assessment cycle can lead to the implementation of a fatigue crack repair by crack removal. To improve the fatigue performance of the crack repair, residual compressive stresses induced by peening can be considered. This paper is in relation to the peening effect estimation on stress intensity factors in fatigue crack repaired weldments, since the stress intensity factor is a key parameter in fracture mechanics calculations. A set of T-butt specimens were experimentally fatigue tested and crack propagation data was gathered for the calculation of stress intensity factors. The experiments were designed to estimate the residual compressive stress depth layer and its effect on crack propagation inhibition. Experimental estimation of the peening effect on stress intensity factors in fatigue crack repaired weldments was validated by comparison against an analytical weight function solution. Experimental stress intensity factors determined from a set of fatigue tested T-butt specimens allowed estimating preliminarily that peening has a limited effect on fatigue crack propagation inhibition for edge repaired T-butt weldments subjected to bending loading.

Analytical reconstruction of isotropic turbulence spectra based on the Gaussian transform

The field of application of the Random Particle Mesh (RPM) method used to simulate turbulence-induced broadband noise in several aeroacoustic applications is improved to realise isotropic turbulence spectra. With this method turbulent fluctuations are synthesised by filtering white noise with a Gaussian filter kernel that in turn gives a Gaussian spectrum. The Gaussian filter is efficient and finds wide-spread applications in stochastic signal processing. However Gaussian spectra do not correspond to real turbulence spectra. Thus in turbo-machines the von Kármán, Liepmann, and modified von Kármán spectra are more realistic model spectra. In this note we analytically derive weighting functions to realise arbitrary isotropic solenoidal spectra using a superposition of weighted Gaussian spectra of different length scales. The analytic weighting functions for the von Kármán, the Liepmann, and the modified von Kármán spectra are derived subsequently. Finally a method is proposed to discretise the problem using a limited number of Gaussian spectra. The effectivity of this approach is demonstrated by realising a von Kármán velocity spectrum using the RPM method.

Weight functions and stress intensity factors for pin‐loaded single‐edge notch bend specimen

AbstractA recently developed pin‐loaded single‐edge notch bend specimen provides an alternative to the single‐edge notch tension specimen commonly used for small‐crack growth testing. In this paper, weight functions for pin‐loaded single‐edge notch bend specimen are derived by using two methods, the classical analytical weight function method and the newly developed numerical weight function complex variable Taylor series expansion method. Excellent agreement between the two methods is achieved. Based on these weight functions, accurate stress intensity factors for two load cases, that is, pin‐loading and Dugdale loading, which is required for plasticity‐induced crack‐closure analysis based on the strip‐yield model, are determined.

Wide-range weight functions and stress intensity factors for arbitrarily shaped crack geometries using complex Taylor series expansion method

The recently developed complex Taylor series expansion method for computing weight functions (WCTSE) of arbitrary 2D crack geometry is further studied. Improvements on several aspects are made to raise solution accuracy. Wide-range weight functions are determined for four different crack geometries. Methods for accuracy assessment using the Green’s function or severe load conditions are proposed. The WCTSE numerical weight function and the conventional analytical weight function approaches are shown to be complementary. It is envisaged that combination of the two approaches will lead to more widespread applications of the weight function methods for fracture and crack growth analyses.

FORWARD PRICES AS FUNCTIONALS OF THE SPOT PATH IN COMMODITY MARKETS MODELED BY LEVY SEMISTATIONARY PROCESSES

We show that the forward price can be represented as a functional of the spot price path in the case of Lévy semistationary (LSS) models for the spot dynamics. The functional is a weighted average of the historical spot price in general, and is derived by means of the Laplace transform. For the special cases of continuous-time autoregressive (CAR) moving average and Gamma-LSS processes for the spot dynamics, we are able to produce an analytical weight function. Both classes of processes are of interest in markets for power, weather and shipping, and we provide a discussion of the results based on numerical examples.

An Inverted Preisach Model With Analytical Weight Function and Its Numerical Discrete Formulation

This paper proposes a numerical discrete formulation for an inverted hysteresis model, which is based on the classic Preisach model, but applies an inverted switch operator. The inverted switch operator is modified from the switch operator of the classic Preisach model. By applying this inverted switch operator, the inverted model retains the wiping-out and congruency properties. The distribution of the weight function can be approximated with an analytical function. By means of a quasi-Newton method algorithm, the small number of parameters in this function can be directly determined from major loops. Iterative procedure is circumvented in the implementation of this model in finite-element analysis. The simulated hysteresis loops, such as minor loops and reversal curves, show good agreement to the measurement data. To realize efficient numerical calculation, the integral expression of the inverted model is rewritten in a discrete form. The double integration of the weight function is numerically approximated in the precalculation stage. In this way, it costs low memory usage and short calculation time during the calculation procedure.

A Fully Parametric Weight Function for Inclined Kinked Edge Cracks

A general method for evaluating the Stress Intensity Factors (SIFs) of an inclined kinked edge crack in a semi-plane is presented. An analytical Weight Function (WF) with a matrix structure is derived by extending a method developed for a straight inclined edge crack. The effects of the principal geometrical parameters governing the problem were studied through a parametric Finite Element (FE) analysis, carried out for different reference loading conditions. The proposed WF can be used to produce efficient and accurate evaluations of the SIFs for cracks with initial inclination angle in the range [–60°,60°], kink angle in the range [- 90°,90°], and kinked crack to initial crack length ratios in the range [0,1/4].

Residual Stresses and Stress Intensity Factor Calculations in T-Welded Joints

In welded joint, the residual stresses effect can be considered using the residual stress intensity factor (K res). In this study, K res is calculated using the analytic weight function method (WFM) and the polynomial distribution of residual stresses (σ res). The different residual stress distributions have been used analytically. It is to be emphasized that the current approach is little investigated. This is because the weight function has already been developed to calculate K for a crack that had already existed, and hence to calculate the stress distribution and stress intensity factor over the crack face. Therefore, the current approach calculates K res with σ res consideration for the crack which initiates and propagates until failure. The validity to use the proposed weight function has been shown. The results of K res have been compared with those obtained from FEM.

Consideration of the residual stress distributions in fatigue crack growth calculations for assessing welded steel joints

ABSTRACTTo better understand the crack closure and propagation, an analytical model is established. The residual stress effect on fatigue crack growth equations has been considered using the residual stress intensity factor (SIF) (Kres). The joint geometries, residual stress distributions (σres) and residual stress ratio (Rres) were considered also. Kres are calculated using the analytical weight function (WF) method and different residual stress distributions. It is to be emphasized that the current approach is little investigated. This is because the WF has already been developed to calculate SIF for an existing crack. The current approach calculates Kres for the crack that initiates and propagates until failure. Different stress distributions have been used, and Rres is defined. The validity of using the WF has been shown. SIF due to applied load (Kapp) and applied stress ratio (Rapp) have been considered. Fatigue crack growth rate was investigated in accordance with the current approach. The results have been verified and benchmarked.

Stress intensity factors of inclined kinked edge cracks: A simplified approach

A simplified approach for evaluating the stress intensity factors of an inclined edge kinked crack in a semi-plane is presented. This is based on an analytical weight function, recently derived by the authors, accounting for the principal geometrical parameters governing the problem: the initial inclination angle, the kinked angle and the ratio between the length of the two crack segments a/a0. Since the weight function coefficients must be expressed in tabular form parametric in the a/a0 ratio, the computational efficiency of the method was improved by investigating the possibility of substituting the original kinked crack with an equivalent crack connecting the two extremities of the original crack and made up by a main linear segment ending with an infinitesimal kinked part aligned with the terminal kinked segment of the original crack. It was found out that an acceptable stress intensity factors estimation is obtained for crack kinking angles comprised between −30° and 30° and a/a0 ratios not higher than 1/5.

A linearized two-stream radiative transfer code for fast approximation of multiple-scatter fields

Performance is an issue for radiative transfer simulations in hyper-spectral remote sensing backscatter retrieval algorithms. 2-Stream models are often used to speed up flux and radiance calculations. Here we present a linearized 2-stream multiple-scatter code, with the ability to generate analytic weighting functions with respect to any atmospheric or surface property. We examine 2-stream accuracy for the satellite intensity diffuse field, and the corresponding Jacobians for total ozone column and surface albedo, for an application in the ozone UV Huggins bands.

Modeling and measurement of creep- and rate-dependent hysteresis in ferroelectric actuators

AbstractWe present a new model for the simulation of the hysteretic large-signal behavior in ferroelectric materials. The model is based on the Preisach operator and takes advantage of an analytic weight function for the underlying fundamental switching operators. The five independent parameters describing this weight function have been determined for a discoidal piezoceramic actuator by adapting the model output to measurements of the polarization. Since the classical Preisach model is only valid for creep- and rate-independent hysteresis loops, it is inappropriate to describe the entire ferroelectric material behavior. Therefore, we extended our model by an additional drift operator to consider creep phenomena. Furthermore, the rate-dependence of the hysteresis loops is described by a frequency-dependent parameter of the analytic weight function. For the identification and verification of the model, measurements have been performed using two different measurement principles: A modified Sawyer–Tower circuit as well as a method based on the integration of the electrical current. The agreement between measurements and simulations highlights the benefits of the enhanced hysteresis model.