Software Framework for Parameter Updating and Finite-Element Response Sensitivity Analysis

The finite-element software framework OpenSees is extended with parameter updating and response sensitivity capabilities to support client applications such as reliability, optimization, and system identification. Using software design patterns, member properties, applied loadings, and nodal coordinates can be identified and repeatedly updated in order to create customized finite-element model updating applications. Parameters are identified using a Chain of Responsibility software pattern, where objects in the finite-element model forward a parameterization request to component objects until the request is handled. All messages to identify and update parameters are passed through a Facade that decouples client applications from the finite-element domain of OpenSees. To support response sensitivity analysis, the Strategy design pattern facilitates multiple approaches to evaluate gradients of the structural response, whereas the Visitor pattern ensures that objects in the finite-element domain make the proper contributions to the equations that govern the response sensitivity. Examples demonstrate the software design and the steps taken by representative finite-element model updating and response sensitivity applications.     

Handling of Constraints in Finite-Element Response Sensitivity Analysis

In this paper, the direct differentiation method (DDM) for finite-element (FE) response sensitivity analysis is extended to linear and nonlinear FE models with multi-point constraints (MPCs). The analytical developments are provided for three different constraint handling methods, namely: (1) the transformation equation method; (2) the Lagrange multiplier method; and (3) the penalty function method. Two nonlinear benchmark applications are presented: (1) a two-dimensional soil-foundation-structure interaction system and (2) a three-dimensional, one-bay by one-bay, three-story reinforced concrete building with floor slabs modeled as rigid diaphragms, both subjected to seismic excitation. Time histories of response parameters and their sensitivities to material constitutive parameters are computed and discussed, with emphasis on the relative importance of these parameters in affecting the structural response. The DDM-based response sensitivity results are compared with corresponding forward finite difference analysis results, thus validating the formulation presented and its computer implementation. The developments presented in this paper close an important gap between FE response-only analysis and FE response sensitivity analysis through the DDM, extending the latter to applications requiring response sensitivities of FE models with MPCs. These applications include structural optimization, structural reliability analysis, and finite-element model updating.     

Second-Order Sensitivities of Inelastic Finite-Element Response by Direct Differentiation

In this paper analytical equations are developed and implemented to obtain second-order derivatives of finite-element responses with respect to input parameters. The work extends previous work on first-order response sensitivity analysis. Of particular interest in this study is the computational feasibility of obtaining second-order response sensitivities. In the past, the straightforward finite difference approach has been available, but this approach suffers from serious efficiency and accuracy concerns. In this study it is demonstrated that analytical differentiation of the response algorithm and subsequent implementation on the computer provides second-order sensitivities at a significantly reduced cost. The sensitivity results are consistent with and have the same numerical precision as the ordinary response. The computational cost advantage of the direct differentiation approach increases as the problem size increases. Several novel implementation techniques are developed in this paper to optimize the computational efficiency. The derivations and implementations are demonstrated and verified with two finite-element analysis examples.     

Sensitivity Statistics of 3D Structures under Parametric Uncertainty

Determination of sensitivity gradient is a major prerequisite for structural optimization, reliability assessment, and parameter identification. As the conventional deterministic sensitivity analysis cannot provide complete information, stochastic analysis is needed to tackle the uncertainties in structural parameters. This study focuses on the utility of the stochastic finite-element method for random response sensitivity analysis. The stochastic modeling of a random parameter is based on a commonly used 2D local averaging method generalized for a 3D case. The Choleski decomposition technique is then employed for digital simulation. The Neumann expansion based finite-element simulation method is extended for stochastic sensitivity analysis. This technique leads to a considerable saving of computational time. Example problems are used to compare the accuracy of this method to the direct Monte Carlo simulation and perturbation method in terms of varying stochasticity and efficiency in CPU time.     

Nonlinear Finite-Element Analysis Software Architecture Using Object Composition

Object composition offers significant advantages over class inheritance to develop a flexible software architecture for finite-element analysis. Using this approach, separate classes encapsulate fundamental finite-element algorithms and interoperate to form and solve the governing nonlinear equations. Communication between objects in the analysis composition is established using software design patterns. Root-finding algorithms, time integration methods, constraint handlers, linear equation solvers, and degree of freedom numberers are implemented as interchangeable components using the Strategy pattern. The Bridge and Factory Method patterns allow objects of the finite-element model to vary independently from objects that implement the numerical solution procedures. The Adapter and Iterator patterns permit equations to be assembled entirely through abstract interfaces that do not expose either the storage of objects in the analysis model or the computational details of the time integration method. Sequence diagrams document the interoperability of the analysis classes for solving nonlinear finite-element equations, demonstrating that object composition with design patterns provides a general approach to developing and refactoring nonlinear finite-element software.     

Parameter Sensitivity and Importance Measures in Nonlinear Finite Element Reliability Analysis

Finite element reliability methods allow the analyst to define material, load, and geometry parameters as random variables to represent uncertainties in these model parameters. Approximate probabilistic analysis methods produce estimates of the response variance/covariances, probabilities of exceeding specified structural performance thresholds, and parameter importance measures. A necessary ingredient for such analysis is consistent, efficient, and accurate algorithms for computing finite element response sensitivities. In this paper, unified response sensitivity equations with respect to material, load, and geometry parameters are developed for the time- and space-discretized finite element model. The sensitivities with respect to nodal coordinates and global shape parameters in the presence of material and geometric nonlinearities represent an extension of previous work. Practical computer implementation issues are emphasized. The equations are implemented in the comprehensive, open-source, object-oriented finite element software OpenSees. Importance measures from reliability analysis, employing the sensitivity results, are presented to enable the investigation of the relative importance of uncertainty in the parameters of a finite element model. Two example applications demonstrate that the variability in nodal coordinates of a structure can be a significant source of uncertainty along with that in key material and load parameters.     

Finite-Element Model Updating for the Kap Shui Mun Cable-Stayed Bridge

This paper presents the implementation of the finite-element model updating for the Kap Shui Mun Bridge, a 430 m main span double-deck cable-stayed bridge in Hong Kong. The dynamic characteristics of the bridge have been studied through both three-dimensional finite-element prediction and field vibration measurement previously. In this paper, the developed finite-element model is updated based on the field measured dynamic properties. A comprehensive sensitivity study to demonstrate the effects of various structural parameters (including the connections and boundary conditions) on the modes of concern is first performed, according to which a set of structural parameters are then selected for adjustment. The finite-element model is updated in an iterative fashion so as to minimize the differences between the predicted and the measured natural frequencies. The final updated finite-element model for the Kap Shui Mun Bridge is able to produce natural frequencies in good agreement with the measured ones, and can be helpful for a more precise dynamic response prediction.     

Consistent Finite-Element Response Sensitivity Analysis

This paper examines the important issue of response sensitivities of dynamic models of structural systems to both material and (discrete) loading parameters. Plasticity-based finite-element models of structural systems subjected to base excitation such as earthquake loading are considered. The two methods for computing the response sensitivities, namely, (1) discretizing in time the time continuous-spatially discrete response equations and differentiating the resulting time discrete-spatially discrete response equations with respect to sensitivity parameters, and (2) differentiating the time continuous-spatially discrete response equations with respect to sensitivity parameters and discretizing in time the resulting time continuous-spatially discrete response sensitivity equations, are clearly distinguished. The discontinuities in time of the response sensitivities arising due to material state transitions in the plasticity models, and their propagation from the quadrature point level to the global structural response level are discussed using a specific one-dimensional plasticity model. The procedure to obtain the exact sensitivities of the numerical nonlinear finite-element response, including proper capture of their discontinuities, is formalized. Application examples illustrating the concepts are presented at the end.     

Bayesian Modal Updating using Complete Input and Incomplete Response Noisy Measurements

The problem of identification of the modal parameters of a structural model using complete input and incomplete response time histories is addressed. It is assumed that there exist both input error (due to input measurement noise) and output error (due to output measurement noise and modeling error). These errors are modeled by independent white noise processes, and contribute towards uncertainty in the identification of the modal parameters of the model. To explicitly treat these uncertainties, a Bayesian framework is adopted and a Bayesian time-domain methodology for modal updating based on an approximate conditional probability expansion is presented. The methodology allows one to obtain not only the optimal (most probable) values of the updated modal parameters but also their uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed with the updating of a theoretical finite-element model based on these modal estimates. The proposed approach requires only one set of excitation and corresponding response data. It is found that the updated probability density function (PDF) can be well approximated by a Gaussian distribution centered at the optimal parameters at which the posterior PDF is maximized. Numerical examples using noisy simulated data are presented to illustrate the proposed method.     

Probabilistic Stability Analyses of Embankments Based on Finite-Element Method

Conventional limit equilibrium methods are commonly used to assess the stability of embankments. The finite-element method, as an alternative to limit equilibrium methods, is being increasingly used in the deterministic stability analysis of slopes or embankments. In this paper, a practical procedure for integrating the finite-element method and the limit equilibrium methods into probabilistic stability analysis for embankments is presented. The response surface method is adopted to approximate the performance function for the stability problems and the first-order reliability method is used to calculate the reliability index based on an intuitive expanding ellipsoid perspective. The advantages of the response surface method as a bridge between stand-alone numerical packages and spreadsheet-based reliability analysis via automatic constrained optimization are demonstrated and discussed through a hypothetical two-layer slope and an actual case of the James Bay Dykes. The results show the ease and successful implementation of the proposed procedure for reliability analysis of embankments.     

Probability and Sensitivity Analysis in Flotation Circuit of Bama Lead and Zinc Processing Plant Using Monte Carlo Simulation Method

The feed to mineral processing plants has fluctuations in assay, particle size, etc. These fluctuations have a strong influence on the efficiency of purifying lead and zinc processes. In simple circuits of mineral processing units, sensitivity analysis is applicable using analytical solutions, but this method is very difficult, time consuming and inaccurate for simulation of complex circuits. Nevertheless, Monte Carlo simulation method is an accepted tool for probability calculation and sensitivity analysis of complex circuits. In this paper, input data are collected from line 1 of flotation circuit of Bama lead and zinc processing plant (Isfahan, Iran) during 3 months of three 8-hour shifts per day. Distribution and probability density function of input data are determined using @Risk software in Excel. Then, the probability density function of output data (flow rates of lead concentrate, zinc concentrate, tailings, recovery of lead and zinc) are calculated using Monte Carlo simulation (with 100,000 iterations). Finally, the sensitivity of output parameters to input parameters is determined.     

Reliability-Based Assessment of Suspension Bridges: Application to the Innoshima Bridge

Many suspension and cable-stayed bridges were designed and constructed between Honshu Island and Shikoku Island in Japan. All these bridges were designed according to the allowable stress design method. In the allowable stress design method, it is not possible to quantify the reliabilities of both bridge components and the entire bridge system. Therefore, in light of current reliability-based design philosophy, there is an urgent need to assess the safety of suspension bridges from a probabilistic viewpoint. To develop cost-effective design and maintenance strategies, it is necessary to assess the condition of suspension bridges using a reliability-based approach. This is accomplished by a probabilistic finite-element geometrically nonlinear analysis. This study describes an investigation into the reliability assessment of suspension bridges. The combination of reliability analysis and geometrically nonlinear elastic analysis allows the determination of reliabilities of suspension bridges. A probabilistic finite-element geometrically nonlinear elastic code, created by interfacing a system reliability analysis program with a finite-element program, is used for reliability assessment of suspension bridges. An existing suspension bridge in Japan, the Innoshima Bridge, is assessed using the proposed code. The assessment is based on static load effects. Reliabilities of the bridge are obtained by using 2D and 3D geometrically nonlinear models. Furthermore, damage scenarios are considered to assess the effects of failure of various elements on the reliability of undamaged components and on the reliability of the bridge. Finally, sensitivity information is obtained to evaluate the dominant effects on bridge reliability.     

Simulation of macroscopic solidification with an incorporated one-dimensional microsegregation model coupled to thermodynamic software

In this article, a numerical model is presented which predicts phase distributions and dendrite arm spacings for a realistic casting within suitable CPU time. Three software components are coupled to perform calculations: (1) an FEM simulation package for the macroscopic temperature field, (2) an FDM code for the microstructure parameters, and (3) a thermodynamic software package for equilibrium calculations at the interfaces. The macrosoftware provides the micromodule with the present temperature at each node of a finite-element grid. At each such node, the micromodule calculates the dendrite arm spacings, the phase amounts, and the diffusion-controlled segregation profiles for the current time-step using equilibrium information from the thermodynamic software. A change of solid fraction and of the phase concentrations results in the release of latent heat and in the change of the heat capacity. These values are used as input parameters in the macrosoftware for the temperature calculation in the next time-step. Simulations have been performed for the ternary alloy AlCu4Mg1 and the results have been compared to “traditional” temperature calculations and to experimentally determined phase fractions and dendrite arm spacings. Measurements have been done by means of an interactive image analysis system over the entire breadth of an ingot casting at four different heights as well as at three different longitudinal cuts.     

Influence of Spatial Variability on Slope Reliability Using 2-D Random Fields

The paper investigates the probability of failure of slopes using both traditional and more advanced probabilistic analysis tools. The advanced method, called the random finite-element method, uses elastoplasticity in a finite-element model combined with random field theory in a Monte-Carlo framework. The traditional method, called the first-order reliability method, computes a reliability index which is the shortest distance (in units of directional equivalent standard deviations) from the equivalent mean-value point to the limit state surface and estimates the probability of failure from the reliability index. Numerical results show that simplified probabilistic analyses in which spatial variability of soil properties is not properly accounted for, can lead to unconservative estimates of the probability of failure if the coefficient of variation of the shear strength parameters exceeds a critical value. The influences of slope inclination, factor of safety (based on mean strength values), and cross correlation between strength parameters on this critical value have been investigated by parametric studies in this paper. The results indicate when probabilistic approaches, which do not model spatial variation, may lead to unconservative estimates of slope failure probability and when more advanced probabilistic methods are warranted.     

Stochastic Finite-Element Approach to Quantify and Reduce Uncertainty in Pollutant Transport Modeling

One of the important issues in simulation of contaminant transport in the subsurface is how to quantify the hydraulic properties of soil that are randomly variable in space because of soil heterogeneity. Stochastic approaches have the potential to represent spatially variable parameters, making them an appropriate tool to incorporate the effects of the spatial variability of soil hydraulic properties on contaminant fate. This paper presents development and application of a numerical model for simulation of advective and diffusive-dispersive contaminant transport using a stochastic finite-element approach. Employing the stochastic finite-element method proposed in this study, the response variability is reproduced with a high accuracy. Comparison of the results of the proposed method with the results obtained using the Monte?Carlo approach yields a pronounced reduction in the computation cost while resulting in virtually the same response variability as the Monte?Carlo technique.     

Vibration Serviceability of a Building Floor Structure. I: Dynamic Testing and Computer Modeling

Buildings with large column-free floors or long-cantilevered structures can be susceptible to annoying vibrations due to everyday occupants’ activities such as walking. Computer modeling and analytical representation of building structural properties to predict the floor response subjected to excitations due to human activities are important issues that require further studies. Vibration testing and analysis of built structures can assist in more accurate estimation of structure dynamic properties. This paper presents the results of the modal testing conducted on an office building floor and analysis of the collected vibration measurements. It compares these results with the structural response using computer analyses. It also presents a sensitivity study to assess the importance of various structural parameters on the floor dynamic response. From the results presented, it is concluded that for the structure used in this study the raised flooring and nonstructural elements acted mainly as added mass and did not contribute to the floor damping. Conclusions are also made on the importance of various structural parameters on floor response and the analysis of the modal test results.     

Development of Finite-Element Modeling Approach for Lateral Load Analysis of Dry-Glazed Curtain Walls

This paper presents a finite-element modeling option to provide an analytical approach for a seismic analysis of dry-glazed curtain-wall systems. In this modeling approach, Ansys finite-element software was used to model the glass panel, aluminum glazing frame, perimeter rubber gaskets, rubber setting and side blocks, glass-to-frame clearances, and glass-to-frame contact once the clearance was overcome by in-plane drift. The results of the finite-element modeling of the curtain-wall system were compared with full-scale laboratory test results. The effect of some of the parameters such as gasket friction and aspect ratio were evaluated. The study showed that finite-element modeling is a viable approach for analytical evaluation of curtain walls. The modeling can function to predict the drift associated with glass-panel cracking. Further refinement of the modeling approach developed can increase the accuracy of the prediction.