Variability Response Function Research Articles

Reform of the construction of the professional skills course cluster of musicology in colleges and universities in the context of aesthetic immersion action based on the VAR model

Abstract In the context of aesthetic education infiltration action, the construction of musicology majors in colleges and universities needs to design diverse teaching contents. This paper analyzes the dynamic impact of the construction of musicology professional skill course clusters with the help of the variable impulse response function of the VAR model and explores the promotional relationship between aesthetic education and music professional literacy. Aiming at the problem that the VAR model cannot reflect the time-varying features and nonlinear features, the TVP-VAP model is used to capture the time-varying features, and at the same time, the Bayesian algorithm is used to calculate the parameters in the model to improve the operation efficiency of the model. After the design was completed and put into practical use, the professional knowledge scores of musicology majors in colleges and universities and students’ aesthetic ability scores in January-October 2023 were selected as databases to establish the TVP-VAR model. When implementing aesthetic education, the response value is 0. From the second month onwards, aesthetic education will have a lasting positive effect and level off, and the response value is around 0.003. The value of the impact effect of students’ professionalism on aesthetic education is 0, which indicates a lag, and shows a gradual upward trend in periods 1-4, with a maximum response value of 0.26. This paper provides a quantitative analysis of the effects of integrating aesthetic education and music and proposes a new direction for reforming the curriculum of the music major.

Variability response functions for statically indeterminate structures

The response variability of statically indeterminate beam structures caused by uncertain material properties modeled by random fields is studied in this paper. By solving the governing equations of the statically indeterminate structure, the response bending moment along the length of the beam, M(x), is expressed as a function of its (random) zero-moment location. This approach, combined with a second-order Taylor series expansion of the random zero-moment location, leads to novel Variability Response Function-based integral expressions for the variance of the response bending moment, Var[M(x)]. Variability Response Functions have numerous desirable attributes including the capability to perform a full sensitivity analysis of the response variability with respect to the spectral characteristics of the random field modeling the uncertain material/system properties and establishing realizable upper bounds of the response variability. The proposed expressions are fundamentally different from all currently existing expressions that are based inherently on first-order Taylor series expansions, as they involve: (i) two distinct integrals in the expression for Var[M(x)] and, (ii) an expression for the mean value ɛ[M(x)] that depends on the stochastic field modeling the uncertain material properties. In general, the proposed second-order Taylor series-based formulation is expected to provide more accurate results. Extensive numerical examples are provided where the accuracy of the results obtained using the proposed formulation is validated using Monte Carlo simulations involving stochastic fields that follow truncated Gaussian and shifted lognormal probability distribution functions. These Monte Carlo simulation results indicate that the proposed Variability Response Functions are probability-distribution-independent.

Application of Variability Response Function to Cross Laminated Timber

Variation in material properties leads to variation in stiffness, or displacement response, of structural systems. This paper presents an efficient method to calculate the variability in stiffness of a three-layer Cross Laminated Timber (CLT) panel as a result of heterogeneous elastic properties in the constituent lumber. Typically, problems involving stochastic material properties are solved by Monte Carlo simulations, which is computationally expensive for complex systems. The Variability Response Function is a well-established concept for efficient evaluation of the variance of stochastic systems where properties are modeled by random fields. A new method to study the variation in CLT panel stiffness is developed by applying the Variability Response Function to CLT geometry. Using this method, novel CLT panel layups can be efficiently studied, increasing opportunities for traditionally low-value species to be used in CLT panels. Three case studies are provided to demonstrate the application and utility of the developed method.

Variability response function approach for foundation reliability

This paper explores the applicability of variability response functions to nonlinear soil–structure interaction problems, focusing on the impacts of spatially variable soil properties on foundation reliability regarding the settlement response. An estimation scheme is proposed to obtain the response functions, which involves a periodic function to approximate the relationship between foundation response and phase angles representing the soil variability patterns. Using a single set of realizations, the response function approach enables the evaluation of foundation reliability under various spatial variability patterns, including different autocovariance distances in three dimensions and the rotated anisotropy features of soil variations. This leads to significant reductions in computational demands compared to previous attempts of random field modelling, which often involved individual Monte Carlo simulations for each combination of spatial variability parameters. The proposed approach is applied to both shallow foundation and piled foundation cases, illustrating its range of applicability. For linear-elastic systems, the approach is shown to be effective for various coefficients of variation in soil variability. For elastic–plastic pile group analyses, the approach leads to efficient evaluation of the statistics of average and differential settlements of the pile group, both of which compare favourably with conventional random field simulation techniques. Since it does not require multiple random field realizations, the approach is particularly efficient in identifying the worst-case scenario of autocovariance distances that corresponds to the largest uncertainty in foundation response. This can become a useful tool for conservative reliability assessments under a project setting, since site-specific estimates of random field parameters are often imprecise due to limited site data.

Design of on‐farm precision experiments to estimate site‐specific crop responses

AbstractSite‐specific prescriptions require estimating response functions to controllable inputs across the field. The methodology of applying geographically weighted regression to on‐farm precision experimentation studies opens new opportunities to study site‐specific responses to inputs in farmers' fields by locally estimating the regression coefficients. However, the effect of the experiment's spatial layout, such as plot dimensions and randomization, and spatial structure of the yield response on the experiment performance are yet to be studied. Detailed information about these effects is needed to improve trial design to detect site‐specific responses. A simulation study was conducted using 14,400 fields of 37 ha and 9‐m resolution. Coefficients from a spatial variable response function were drawn from five random fields generated by unconditional Gaussian geostatistical simulations. Four levels of nitrogen were assigned to plots using 18 systematic and randomized chessboard designs with different plot sizes. Simulated yield data was obtained by combining the coefficients, treatment, and random error. The effect of spatial structure and the designs was assessed with measures of agreement between the true and estimated maps of regression coefficients. The ability to capture or approximate the true spatial pattern of the response function increased as the underlying response function's spatial structure increases. Overall differences in performance between design were observed across the spatial structure tested, mostly related to randomization and plot dimensions. In general best results were achieved by systematic designs with small or intermediate plot sizes (r = 0.54 ± 0.05, MAE = 0.005 ± 0.0005, SDR = 0.81 ± 0.06, and CP = 0.50 ± 0.04). Our methodology provides a path for testing designs under different spatial variability scenarios.

Dynamic variability response functions for stochastic wave propagation in soils

In this paper, shear wave propagation in soils is examined in a stochastic context considering spatial variability of the shear modulus soil parameter. To this purpose, the recently established concept of dynamic mean and variability response functions (DMRF, DVRF) is reformulated in the framework of stochastic finite element analyses of shear wave propagation problems in order to efficiently calculate the response time history statistics of the soil surface. Similarly to the approximation formulas of classical VRFs, a fast Monte Carlo simulation procedure is implemented to numerically evaluate the above functions in the time domain. The main advantage of the proposed methodology lies on the independence of the DMRF and DVRF on the marginal probability density function and correlation structure of the stochastic system parameter, which in our case is assumed to be the inverse of the soil shear modulus 1/G. By integrating the product of the spectral density of 1/G with the DMRF and DVRF, the mean and variance of the ground response are obtained at each time step of the dynamic analysis. The method also allows for the estimation of time dependent but spectral and probability distribution free upper bounds of the response mean and variance. To illustrate the efficiency and applicability of the proposed approach, stochastic finite element analyses of wave propagation of a Ricker synthetic wavelet as well as a recorded earthquake motion in 1D and 2D soil domains are performed and a sensitivity analysis is carried out with respect to various correlation structures of the underlying random fields representing 1/G. The accuracy of the proposed methodology is validated with comparison to direct Monte Carlo simulation. Useful conclusions regarding the sensitivity of the system response to the spectral characteristics of the underlying random fields representing 1/G are drawn.

Robust design with Variability Response Functions; an alternative approach

In this work a different Robust Design Optimization (RDO) approach is proposed implementing the concept of Variability Response Function (VRF), which is a function that when combined with the power spectral density of the stochastic field that models the system’s uncertainty, formulates an integral expression for the variance of the system’s response. The basic idea is to exploit a very well-known property of the VRF, which is its independence of the stochastic system parameters, in order to obtain global optima that depend only on the deterministic parameters of the problem. This way, optimal structural designs are achieved which are globally insensitive to uncertainties, that is to say they are free of the spectral-distribution characteristics of the stochastic fields modeling the uncertainties. This is achieved by setting in addition to the total material cost, the maximum VRF value as an objective function. The advantages of using the proposed methodology over traditional Robust Design Optimization are illustrated through an application to a frame-type structure where it is demonstrated that the designs achieved through classical RDO for a given stochastic field description are not optimal for a variation on the spectral properties of the random field modeling the system uncertainty, while optimal designs obtained with the VRF-based RDO remain optimum for the worst case scenario stochastic fields.

Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.

Structural response variability under spatially dependent uncertainty: Stochastic versus interval model

This paper deals with structural response variability under spatially varying uncertainties represented using both probabilistic and non-probabilistic models. Attention is focused on Euler–Bernoulli beams with uncertain flexibility subjected to deterministic static loads. Within the probabilistic framework, the uncertain flexibility is modeled as a random field and the well-established concept of variability response function is adopted to derive spectral- and probability-distribution-free upper bounds on the response variability. Within the non-probabilistic context, the uncertain property is represented resorting to a recently proposed interval field model able to quantify the dependency between adjacent values of an interval uncertainty that cannot differ as much as values that are further apart. For statically determinate beams, the interval displacement field and the associated bounds are evaluated analytically by using a Green’s function formulation. Conversely, for statically indeterminate beams, approximate explicit expressions of the bounds of the interval response are derived by applying a finite difference based procedure.Numerical results present consistent comparisons between response variability under random and interval uncertainty for both statically determinate and indeterminate beams.

An adaptive spectral Galerkin stochastic finite element method using variability response functions

SummaryA methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.

Transient response of stochastic finite element systems using Dynamic Variability Response Functions

In this study a methodology is presented for effective analysis of dynamic systems with stochastic material properties. The concept of dynamic mean and variability response functions, recently established for linear stochastic single degree of freedom oscillators, is extended to general finite element systems such as statically indeterminate beam/frame structures and plane stress problems, leading to closed form integral expressions for their dynamic mean and variability response. The integrand of these integral expressions involves the spectral density function of the uncertain material properties and the so called dynamic mean and variability response functions respectively, which are assumed to be deterministic, i.e. independent of the power spectrum as well as the marginal pdf of the uncertain parameters. A finite element method-based fast Monte Carlo simulation procedure is used for the accurate and efficient numerical evaluation of these functions. In order to demonstrate the validity of the proposed procedure, the results obtained using the aforementioned integral expressions are compared to brute-force Monte Carlo simulation. As a further validation of the assumption of independence of the variability response function to the stochastic parameters of the problem, the concept of the generalized variability response function was applied and compared to the steady state dynamic variability response function. The methodology is applied in a dynamically loaded statically indeterminate beam/frame structure and a plane stress problem. The dynamic mean and variability response functions, once established, can be used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the dynamic mean and variance of the response, at practically no additional computational cost.

Generalized variability response functions for two-dimensional elasticity problems

The ability to determine probabilistic characteristics of response quantities in structural mechanics (e.g. displacements, stresses) as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters. The concept of the variability response function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). Proofs have been provided for the existence of VRFs for linear and some nonlinear statically determinate beams. For statically indeterminate structures, the Monte Carlo based generalized variability response function (GVRF) methodology has been proposed recently as a generalization of the VRF concept to indeterminate linear and some nonlinear beams. The methodology computes GVRFs, which are analogous to VRFs for statically determinate structures, and evaluates their dependence (or lack thereof) on the PDF and SDF of the random field, thereby providing an estimate of the accuracy of the GVRF. In this paper, the GVRF methodology is extended to problems involving two-dimensional, linear continua whose stochasticity is characterized by statistically homogeneous random fields. After detailing the GVRF methodology for two-dimensional random fields, two numerical examples are provided: GVRFs are computed for the displacement response and for the effective compliance of linear plane stress systems.

Stochastic variability of effective properties via the generalized variability response function

Homogenization of randomly heterogeneous material properties into effective properties is an essential procedure in facilitating the analysis of a wide range of mechanics problems. Although formulas exist to calculate deterministic effective properties for structures larger than the representative volume element (RVE), no general method other than Monte Carlo simulation exists to evaluate the variability of these effective properties for structures smaller than the RVE. In a recent paper [1], a method was proposed for evaluating the stochastic variability of effective properties by incorporating the variability response function (VRF) concept. Subsequently, the existence of the VRF for effective properties for linear, statically determinate structures was formally proven. The concept of the VRF has been proposed as a means to systematically capture the effect of the spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. Although the existence of VRFs can be formally proven only for statically determinate structures, a Monte Carlo-based methodology has been proposed recently as a generalization of the VRF concept [19]. In this paper, this methodology is extended to establish estimates of the VRF for effective properties of statically indeterminate beams.

Generalized Variability Response Functions for Beam Structures with Stochastic Parameters

A Monte Carlo–based methodology is introduced as a generalization of the variability response function (VRF) concept, applicable to both statically determinate and indeterminate beam structures with possibly large stochastic variations of parameters (bending stiffness or flexibility). This new methodology overcomes all limitations associated with the Taylor expansion-based VRFs used in the past. Two generalized VRFs (GVRFs) result from this methodology: a deflection GVRF and a bending moment GVRF. Numerical evidence indicates that these GVRFs are neither unique nor completely independent of the probabilistic characteristics of the random field modeling the variations of the bending flexibility. The GVRFs are found to be mildly sensitive to the non-Gaussian marginal distribution of this field, but are minimally dependent on its spectral density function. Taking advantage of this finding, a fast Monte Carlo–based methodology for estimating representative GVRFs is also introduced, significantly reducing the computational effort.

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.

Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.

Variability response functions for effective material properties

The variability response function ( VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.

Analysis of mean and mean square response of general linear stochastic finite element systems

A general finite element-based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields. A flexibility-based formulation is followed that does not involve any approximations. Closed-formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper. These integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function (MRF) and the variability response function for the mean square response (VRF 1). The MRF and the VRF 1 have many similarities with the classical variability response function (VRF) used to estimate the variance of the response displacement. The existence of the MRF and the VRF 1 depends on a conjecture that is validated numerically using a brute force Monte Carlo simulation approach. A so-called finite element method-based fast Monte Carlo simulation procedure (FEM-FMCS) is introduced for the accurate and efficient numerical evaluation of the MRF and the VRF 1. This methodology is established for the analysis of stochastic beam/frame structures, as well as for more general stochastic finite element systems (i.e., plane stress and shell problems). Numerical examples are provided including a statically indeterminate beam, a plane stress problem and a shell structure. The MRF and VRF 1 are used to perform sensitivity/parametric analyses with respect to various probabilistic characteristics involved in the problem (i.e., correlation distance, standard deviation) and to establish realizable upper bounds on the mean value of the response displacement. Throughout this paper, the VRF is computed in parallel to the MRF and the VRF 1. It should be noted that although the concept of the VRF has been introduced in an earlier work, this is the first time that a flexibility-based approach has been used to estimate the VRF for two-dimensional problems.

Seasonal response function for daily streamflow investigation

In this study, a linear model with variable transfer response function has been applied for daily streamflow investigation purposes. The reason for a time-varying approach has to be found in the non-linear features of the rainfall-runoff transformation, with seasonal variability affecting model performance when a linear approach is used to describe it. Depending on the number of seasons which can be identified within the hydrologic year, a time-varying response function would require a number of model parameters greater than in the case of a constant response function. Model parameters along with other difficulties related to the identification procedure, the estimation procedure, the robustness of the solutions and the parameters conceptual meaning are to be considered. A general procedure to identify seasonally variable parameters, generating a seasonal variable response function, is proposed to overcome such difficulties. We refer to a recently proposed linear shot noise approach and to a few case studies, showing the improved model descriptive capabilities and the possibility to investigate relationships between climate characteristics and model parameters variability.

Response variability of stochastic frame structures using evolutionary field theory

The integral form for the variance of the response of stochastic statically indeterminate structural systems involving the so-called variability response function (VRF) and the spectral density function of the stochastic field modelling the uncertain system properties is established for the first time in this paper using evolutionary spectra theory. The VRF is a function depending on deterministic parameters related to the geometry, boundary conditions, (mean) material properties and loading of the structural system. No approximations are involved in the derivation of the integral form. However, a conjecture has to be made that is validated using Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). Closed-form expressions can be derived in principle for the VRF of any statically determinate or indeterminate frame system using a flexibility-based formulation. Alternatively, a fast Monte Carlo simulation approach is provided to numerically evaluate the VRF. It is shown in closed-form and numerically that the VRF for statically indeterminate structures is a function of the standard deviation σ ff of the stochastic field modeling the inverse of the elastic modulus. Although the VRF depends on σ ff , it appears to be independent of the functional form of the spectral density function of the stochastic field modeling the uncertain system properties. For statically determinate structures, the VRF is independent of σ ff . The integral form can be used to compute the variance of the system response as well as its upper bound with minimal computational effort. It also provides an excellent insight into the mechanisms controlling the response variability. The upper bounds for the response variance are spectral- and probability-distribution-free requiring knowledge of only the variance of the inverse of the elastic modulus. The proposed bounds are realizable in the sense that it is possible to determine the probabilistic characteristics of the stochastic field that produces them. Several numerical examples are provided demonstrating the capabilities of the methodology.