روش مستقیم تمایز برای تجزیه و تحلیل حساسیت واکنش از محدوده مدل خاک پلاستیسیته سطح

Finite element (FE) response sensitivity analysis is an important component in gradient-based structural optimization, reliability analysis, system identification, and FE model updating. In this paper, the FE response sensitivity analysis methodology based on the direct differentiation method (DDM) is applied to a bounding surface plasticity material model that has been widely used to simulate nonlinear soil behavior under static and dynamic loading conditions. The DDM-based algorithm is derived and implemented in the general-purpose nonlinear finite element analysis program OpenSees. The algorithm is validated through simulation of the nonlinear cyclic response of a soil element and a liquefiable soil site at Port Island, Japan, under earthquake loading. The response sensitivity results are compared and validated with those obtained from Forward Finite Difference (FFD) analysis. Furthermore, the results are used to determine the relative importance of various soil constitutive parameters to the dynamic response of the system. The DDM-based algorithm is demonstrated to be accurate and efficient in computing the FE response sensitivities, and has great potential in the sensitivity analysis of nonlinear dynamic soil-structure systems.

Finite element (FE) response sensitivity analysis is an essential ingredient of gradient-based optimization methods and is required in structural optimization, system identification, reliability, and FE model updating [1], [2], [3] and [4]. Furthermore, the sensitivity analysis results may be used to propagate the material and loading uncertainty to the structural responses of interest. In addition, FE response sensitivities provide invaluable insight into the effects of system parameters on, and their relative importance to, the system response [5]. Several methods are available for response sensitivity analysis, including the Finite Difference Method (FDM), the Adjoint Method (AM), the Perturbation Method (PM), and the Direct Differentiation Method (DDM) [6], [7], [8], [9], [10] and [11]. The FDM is the simplest method for response sensitivity computation, but is computationally expensive and can be negatively affected by numerical noise. The AM is efficient for linear and non-linear elastic systems, but is not a competitive method for path-dependent (i.e., inelastic) problems. The PM is computationally efficient but generally not very accurate. The DDM, on the other hand, is a general, accurate and efficient method that is applicable to any material constitutive model. The DDM-based response sensitivity analysis methodology shows great promise in the analysis of large and complex structural or geotechnical systems. However the DDM method requires analytical derivations and their computer implementation to differentiate the system responses with respect to sensitivity parameters. Over the past decade, the DDM-based sensitivity analysis method has been actively developed and implemented in an open source FE analysis framework known as OpenSees [12]. The DDM has been developed for various constitutive models including uniaxial materials, three-dimensional J2 plasticity models and pressure-independent multi-yield surface J2 plasticity models [13]. These models can be used to simulate truss and beam components in structures, and nonlinear clay behaviors. Detailed descriptions of the DDM-based sensitivity analysis methodology implemented in OpenSees can be found in the literature [14], [15], [16] and [17]. Yet, the method has not been formulated for sandy soils, which usually exhibit different behavior from clayey soils, such as pressure-dependent cyclic behaviors, shear-induced volumetric dilation and contraction, as well as liquefaction under low effective confinement. The objective of this paper is to extend the DDM-based sensitivity analysis to a class of bounding surface models for sandy soils. The bounding surface model has been widely used and proven to be an effective and robust model to simulate the behaviors of sandy materials under cyclic and seismic loading conditions [18], [19], [20] and [21]. The DDM-based sensitivity algorithm is particularly efficient for strongly nonlinear, large-scale problems with a large number of sensitivity parameters. Geotechnical problems modeled using the bounding surface model are such examples. Thus developing a DDM-based sensitivity algorithm for the bounding surface model will allow us to solve a large number of challenging geotechnical problems, such as the earthquake-induced liquefaction phenomenon in sandy soils. When combined with the existing sensitivity analysis framework for clayey soils and soil-structure systems, the DDM-based sensitivity analysis may be readily applied to real soil-foundation-structure interaction systems [17]. This paper provides a summary of the bounding surface model and detailed DDM formulation, followed by examples to validate the DDM-based FE response sensitivity algorithm. The algorithm is applied to study the sensitivity of liquefied ground responses observed at Port Island in Japan under a real earthquake scenario. The results are further used to identify the relative importance of the soil parameters to the ground surface response.

The DDM method is a general, accurate and efficient method for computing FE response sensitivities to model parameters, especially in the case of nonlinear materials. This paper applies the DDM-based response sensitivity analysis methodology to a bounding surface plasticity material model that has been widely used to simulate nonlinear sandy soil behaviors under static and dynamic loading conditions. The algorithm is implemented in the general-purpose nonlinear FE analysis software Open Sees. The new algorithm and its software implementation are validated through two application examples, in which the DDM-based response sensitivities are compared with their counterparts obtained using FFD analysis. The advantage of the DDM method over the FFD method is also highlighted through convergence tests. In the application example, the normalized response sensitivity analysis results are also used to measure the relative importance of the soil constitutive parameters in regards to the ground surface displacement and acceleration in the case of ground liquefaction. The example illustrates the use of finite element response sensitivity analysis to determine the relative importance of material parameters for specified system response parameters. The work presented in this paper significantly broadens the application of DDM-based response sensitivity analysis, since it enables numerous applications involving the use of the bounding surface plasticity material model. Work is underway to extend the present study to sensitivity analysis of large-scale nonlinear soil-structure interaction systems.