Uncertainty quantification of inverse analysis for geomaterials using probabilistic programming

Uncertainty is an essentially challenging for safe construction and long-term stability of geotechnical engineering.The inverse analysis is commonly utilized to determine the physico-mechanical parameters.However,conventional inverse analysis cannot deal with uncertainty in geotechnical and geological systems.In this study,a framework was developed to evaluate and quantify uncertainty in inverse analysis based on the reduced-order model(ROM)and probabilistic programming.The ROM was utilized to capture the mechanical and deformation properties of surrounding rock mass in geomechanical problems.Probabilistic programming was employed to evaluate uncertainty during construction in geotechnical engineering.A circular tunnel was then used to illustrate the proposed framework using analytical and numerical solution.The results show that the geomechanical parameters and associated uncertainty can be properly obtained and the proposed framework can capture the mechanical behaviors under uncertainty.Then,a slope case was employed to demonstrate the performance of the developed framework.The results prove that the proposed framework provides a scientific,feasible,and effective tool to characterize the properties and physical mechanism of geomaterials under uncertainty in geotechnical engineering problems.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

Inverse reliability analysis and design for tunnel face stability considering soil spatial variability

The traditional deterministic analysis for tunnel face stability neglects the uncertainties of geotechnical parameters,while the simplified reliability analysis which models the potential uncertainties by means of random variables usually fails to account for soil spatial variability.To overcome these limitations,this study proposes an efficient framework for conducting reliability analysis and reliability-based design(RBD)of tunnel face stability in spatially variable soil strata.The three-dimensional(3D)rotational failure mechanism of the tunnel face is extended to account for the soil spatial variability,and a probabilistic framework is established by coupling the extended mechanism with the improved Hasofer-Lind-Rackwits-Fiessler recursive algorithm(iHLRF)as well as its inverse analysis formulation.The proposed framework allows for rapid and precise reliability analysis and RBD of tunnel face stability.To demonstrate the feasibility and efficacy of the proposed framework,an illustrative case of tunnelling in frictional soils is presented,where the soil's cohesion and friction angle are modelled as two anisotropic cross-correlated lognormal random fields.The results show that the proposed method can accurately estimate the failure probability(or reliability index)regarding the tunnel face stability and can efficiently determine the required supporting pressure for a target reliability index with soil spatial variability being taken into account.Furthermore,this study reveals the impact of various factors on the support pressure,including coefficient of variation,cross-correlation between cohesion and friction angle,as well as autocorrelation distance of spatially variable soil strata.The results also demonstrate the feasibility of using the forward and/or inverse first-order reliability method(FORM)in high-dimensional stochastic problems.It is hoped that this study may provide a practical and reliable framework for determining the stability of tunnels in complex soil strata.

Three-dimensional reconstruction of precession warhead based on multi-view micro-Doppler analysis

The warhead of a ballistic missile may precess due to lateral moments during release. The resulting micro-Doppler effect is determined by parameters such as the target's motion state and size. A three-dimensional reconstruction method for the precession warhead via the micro-Doppler analysis and inverse Radon transform(IRT) is proposed in this paper. The precession parameters are extracted by the micro-Doppler analysis from three radars, and the IRT is used to estimate the size of targe. The scatterers of the target can be reconstructed based on the above parameters. Simulation experimental results illustrate the effectiveness of the proposed method in this paper.

An Effective Meshless Approach for Inverse Cauchy Problems in 2D and 3D Electroelastic Piezoelectric Structures

In the past decade,notable progress has been achieved in the development of the generalized finite difference method(GFDM).The underlying principle of GFDM involves dividing the domain into multiple sub-domains.Within each sub-domain,explicit formulas for the necessary partial derivatives of the partial differential equations(PDEs)can be obtained through the application of Taylor series expansion and moving-least square approximation methods.Consequently,the method generates a sparse coefficient matrix,exhibiting a banded structure,making it highly advantageous for large-scale engineering computations.In this study,we present the application of the GFDM to numerically solve inverse Cauchy problems in two-and three-dimensional piezoelectric structures.Through our preliminary numerical experiments,we demonstrate that the proposed GFDMapproach shows great promise for accurately simulating coupled electroelastic equations in inverse problems,even with 3%errors added to the input data.

Rockfill material uncertainty inversion analysis of concrete-faced rockfill dams using stacking ensemble strategy and Jaya optimizer

Numerical simulation of concrete-faced rockfill dams(CFRDs)considering the spatial variability of rockfill has become a popular research topic in recent years.In order to determine uncertain rockfill properties efficiently and reliably,this study developed an uncertainty inversion analysis method for rockfill material parameters using the stacking ensemble strategy and Jaya optimizer.The comprehensive implementation process of the proposed model was described with an illustrative CFRD example.First,the surrogate model method using the stacking ensemble algorithm was used to conduct the Monte Carlo stochastic finite element calculations with reduced computational cost and improved accuracy.Afterwards,the Jaya algorithm was used to inversely calculate the combination of the coefficient of variation of rockfill material parameters.This optimizer obtained higher accuracy and more significant uncertainty reduction than traditional optimizers.Overall,the developed model effectively identified the random parameters of rockfill materials.This study provided scientific references for uncertainty analysis of CFRDs.In addition,the proposed method can be applied to other similar engineering structures.

PERTURBATION ANALYSIS OF A CLASS OF MATRIX INVERSE PROBLEMS

A class of matrix inverse problems minimizing ‖A-‖ F on the linear manifold l A=｛A∈R n×m ｜‖AX-B‖ F=min｝ is considered. The perturbation analysis of the solution to these problems is carried out. The perturbation upper bounds of the solution are given for both the consistent and inconsistent cases. The obtained preturbation upper bounds are with respect to the distance from the perturbed solution to the unperturbed manifold.

Modified Lorentz Transformations and Space-Time Splitting According to the Inverse Relativity Model

Analysis of a four-dimensional displacement vector on the fabric of space-time in the special or general case into two Four-dimensional vectors, according to specific conditions leads to the splitting of the total fabric of space-time into a positive subspace-time that represents the space of causality and a negative subspace-time which represents a space without causality, thus, in the special case, we have new transformations for the coordinates of space and time modified from Lorentz transformations specific to each subspace, where the contraction of length disappears and the speed of light is no longer a universal constant. In the general case, we have new types of matric tensor, one for positive subspace-time and the other for negative subspace-time. We also find that the speed of the photon decreases in positive subspace-time until it reaches zero and increases in negative subspace-time until it reaches the speed of light when the photon reaches the Schwarzschild radius.

Inverse analysis determining interfacial properties between metal film and ceramic substrate with an adhesive layer

In the present study, peel tests and inverse analysis were performed to determine the interracial mechanical parameters for the metal film/ceramic system with an epoxy interface layer between film and ceramic. Al films with a series of thicknesses between 20 and 250 μm and three peel angles of 90°, 135° and 180° were considered. A finite element model with the cohesive zone elements was used to simulate the peeling process. The finite element results were taken as the training data of a neural network in the inverse analysis. The interracial cohesive energy and the separation strength can be determined based on the inverse analysis and peel experimental result

An Inverse Analysis of the Comprehensive Medium Parameters and a Simulation of the Crustal Deformation of the Qinghai-Tibet Plateau

Based on the theory of finite element analysis, an inverse analysis model for the comprehensive medium parameters of the Qinghai-Tibet Plateau is set up. With the help of GPS velocity field, the comprehensive crustal medium parameters of the plateau are inversely analyzed and the characteristics of the related movement macroscopically simulated. It is then concluded that the tectonic deformation of the plateau is mainly in the form of a N-S compression accompanied by an E-W stretching, and the present tectonic setting of the plateau should be the result of the collision between the Indian and the Eurasian continents during the Cenozoic.

An efficient probabilistic design approach for tunnel face stability by inverse reliability analysis

In order to maintain the safety of underground constructions that significantly involve geo-material uncertainties,this paper delivers a new computation framework for conducting reliability-based design(RBD)of shallow tunnel face stability,utilizing a simplified inverse first-order reliability method(FORM).The limit state functions defining tunnel face stability are established for both collapse and blow-out modes of the tunnel face failure,respectively,and the deterministic results of the tunnel face support pressure are obtained through three-dimensional finite element limit analysis(FELA).Because the inverse reliability method can directly capture the design support pressure according to prescribed target reliability index,the computational cost for probabilistic design of tunnel face stability is greatly reduced.By comparison with Monte Carlo simulation results,the accuracy and feasibility of the proposed method are verified.Further,this study presents a series of reliability-based design charts for vividly understanding the limit support pressure on tunnel face in both cohesionless(sandy)soil and cohesive soil stratums,and their optimal support pressure ranges are highlighted.The results show that in the case of sandy soil stratum,the blowout failure of tunnel face is extremely unlikely,whereas the collapse is the only possible failure mode.The parametric study of various geotechnical uncertainties also reveals that ignoring the potential correlation between soil shear strength parameters will lead to over-designed support pressure,and the coefficient of variation of internal friction angle has a greater influence on the tunnel face failure probability than that of the cohesion.

Generalized Inverse Analysis on the Domain ?(A, A^+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B^+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B^+(E, F). In this paper we introduce an unbounded domain ?(A, A^+) in B(E, F) for A ∈ B^+(E, F) and A^+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A^+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A^+)^(-1) is the generalized inverse of T with R(B)=R(A^+) and N(B)=N(A^+), for T∈?(A, A^+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A^+,T) from ?(A,A^+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A^+) : R(T)∩ N(A^+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A^+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A^+). It seems to be useful for developing global analysis and geomatrical method in differential equations.

Analysis and Inverse Substructuring Computation on Dynamic Quality of Mechanical Assembly

Mechanical assembly has its own dynamic quality directly affecting the dynamic quality of whole product and should be considered in quality inspection and estimation of mechanical assembly. Based on functional relations between dynamic characteristics involved in mechanical assembly, the effects of assembling process on dynamic characteristics of substructural components of an assembly system are investigated by substructuring analysis. Assembly-coupling dynamic stiffness is clarified as the dominant factor of the effects and can be used as a quantitative measure of assembly dynamic quality. Two computational schemes using frequency response functions（FRFs） to determine the stiffness are provided and discussed by inverse substructuring analysis, including their applicable conditions and implementation procedure in application. Eigenvalue analysis on matrix-ratios of FRFs before and after assembling is employed and well validates the analytical outcomes and the schemes via both a lumped-parameter model and its analogic experimental counterpart. Applying the two schemes to inspect the dynamic quality provides the message of dynamic performance of the assembly system, and therefore improves conventional quality inspection and estimation of mechanical assembly in completeness.

Inverse analysis of laboratory data and observations for evaluation of backward erosion piping process

An inverse analysis procedure has been developed to interpret collected pore pressure data and observations during backward erosion piping(BEP)initiation and progression in sandy soils.The procedure has been applied to laboratory models designed to mimic the initiation and progression of BEP through a constricted vertical outlet.The inverse analysis uses three-dimensional(3D)finite element method(FEM)to successively produce models of the hydraulic head regime surrounding progressive stages of BEP based on observations at the sample surface and pore pressure measurements obtained from the laboratory models.The inverse analysis results in a series of 3D contour plots that represent the hydraulic-head regime at each stage of the BEP development,allowing for assessing the development of BEP mechanism as well as calculating the critical hydraulic conditions required for various BEP stages to initiate and progress.Interpretation of the results identified four significant stages of the piping process:(1)loosened zone initiation,(2)channel initiation and progression,(3)riser sand fluidization,and(4)loosened zone progression.Interpretation of the hydraulic head contour plots allows assessment of the critical hydraulic gradients needed to initiate and progress various components of the BEP development.

Inverse displacement analysis of the general six degree-of-freedom serial robot based on optimization method

The paper presents a new solution of inverse displacement analysis of the general six degree-of-freedom serial robot.The inverse displacement analysis of the general serial robot is transformed into a minimization problem and then the optimization method is adopted to solve the nonlinear least squares problem with the analytic form of new Jacobian matrix.In this way,joint variables of the general serial robot can be searched out quickly under the desired precision when positions of the three non-collinear end effector points are given.Compared with the general Newton iterative method,the proposed algorithm can search out the solution when the robot is at the singular configuration and the initial configuration used in the optimization method may also be the singular configuration.So the convergence domain is bigger than that of the general Newton iterative method.Another advantage of the proposed algorithm is that positions of the three non-collinear end effector points are usually much easier to be measured than the orientation of the end effector.The inverse displacement analysis of the general 6R(six-revolute-joint) serial robot is illustrated as an example and the simulation results verify the efficiency of the proposed algorithm.Because the three non-collinear points can be selected at random,the method can be applied to any other types of serial robots.

Wedge Splitting Test and Inverse Analysis on Fracture Behaviour of Fiber Reinforced and Regular High Performance Concretes

The fracture behaviour of three fiber reinforced and regular HPC （high performance concretes） is presented in this paper. Two mixes are based on optimization of HPC whereas the third mix was a commercial mix developed by CONTEC ApS （Denmark）. The wedge splitting test setup with 48 cubical specimens was used experimentally and the cracked non-linear hinge model based on the fictitious crack model was applied for the interpretation of the results. The stress-crack opening relationships were extracted by using inverse analysis algorithm for various multi-linear softening curves. This showed that the refinement of the softening curves reflects in improved accuracy of the WST （wedge splitting test） simulation in comparison with bi-linear softening curves with acceptable increase of computational time. Furthermore, the fracture mechanics parameters such as COD （crack opening displacement）, fracture energy and characteristic length were experimentally determined. Experiments were performed at 1, 3, 7 and 28 days. Fracture energy, Gf, was found to increase with age, while the characteristic length, Lch, was found to decrease.

Preliminary analysis of reactivity reconstruction capability based on inverse kinetics method under different initial reactivity states

Online reactivity monitoring plays an important role in operation and safety analyses of fission reactor systems. The inverse kinetics method, which is based on a point kinetics model, is the most widely used method for reactivity reconstruction of critical water reactors. However, this method is seldom applied to the reactivity reconstruction of subcritical reactors. In this study, an inverse kinetics method was employed for the reactivity reconstruction of a lead-based reactor under different initial reactivity states(ρ_0= 0,-2786,-5486,-8367, and-12,371 pcm). The results showed that the deviation in the reactivity of the lead-based subcritical reactor was greater when ρ_0 became smaller. The reactivity reconstructed using the inverse kinetics method was globally underestimated. At a given reactivity perturbation, the relative and absolute errors increased with the decrease in the initial reactivity. At a given initial reactivity, with the increase in the reactivity perturbation, the absolute error increased, whereas the relative error remained the same.This deviation is due to the variation in the external neutron source, spatial-spectral effects, and sub-diffusive effects, which require further study.

Sensitivity and inverse analysis methods for parameter intervals

This paper proposes a sensitivity analysis method for engineering parameters using interval analyses.This method substantially extends the application of interval analysis method.In this scheme,parameter intervals and decision-making target intervals are determined using the interval analysis method.As an example,an inverse analysis method for uncertainty is presented.The intervals of unknown parameters can be obtained by sampling measured data.Even for limited measured data,robust results can also be obtained with the inverse analysis method,which can be intuitively evaluated by the uncertainty expressed in terms of an interval.For complex nonlinear problems,an iteratively optimized inverse analysis model is proposed.In a given set of loose parameter intervals,all the unknown parameter intervals that satisfy the measured information can be obtained by an iteratively optimized inverse analysis model.The influences of measured precisions and the number of parameters on the results of the inverse analysis are evaluated.Finally,the uniqueness of the interval inverse analysis method is discussed.

Inverse Heat Conduction and Data-Type Issues for Advanced Diagnostic Analysis

It is well-known that the local heating/cooling rate implicitly drives the quality and stability of the projection associated with an inverse heat conduction analysis. However, contemporary studies involving inverse heat conduction for heat treatment processes and defense related applications normally use either temperature or strain data. An illustration is presented indicating the merit of using a device capable of either measuring or deducing the heating/cooling rate. Armed with this knowledge certainly suggests that the direct measurement or deduction of the heating/cooling rate can provide a means for developing an alternative and potentially more accurate projection than based on temperature data. A brief theoretical description is offered illustrating that a first-order thermocouple compensation model used in conjunction with an inverse analysis produces a representative prediction of the heating/cooling rate. The presented analysis also offers additional insight into the choice of optimal time-series truncation used by quasi-analytic methods. Using the compensation model, the heating/cooling rate of the desired surface is obtained using the "best" interpretation of the collected thermocouple data. A mathematical formalism is developed and some representative results are presented. Finally, a new mathematical discovery that allows for the time-local temperature errors to be estimated with a high degree of accuracy is briefly elucidated.

A Reinterpretation of Historic Aquifer Tests of Two Hydraulically Fractured Wells by Application of Inverse Analysis, Derivative Analysis, and Diagnostic Plots

Aquifer test methods have greatly improved in recent years with the advent of inverse analysis, derivative analysis, and diagnostic plots. Updated analyses of past aquifer tests allow for improved interpretations of the data to enhance the knowledge and the predictive capabilities of the flow system. This work thoroughly reanalyzes a series of pre- and post-hydraulic fracturing, single-well aquifer tests conducted in two crystalline rock wells in New Hampshire as part of an early 1970’s study. Previous analyses of the data had relied on older manual type-curve methods for predicting the possible effects of hydraulic fracturing. This work applies inverse analysis, derivative analysis, and diagnostic plots to reanalyze the 1970’s aquifer test data. Our results demonstrate that the aquifer tests were affected by changes in flow regimes, dewatering of the aquifer and discrete fractures, and changes due to well development. Increases in transmissivities are related to well development prior to hydraulic fracturing, propagation of a single, vertical fracture hydraulically connecting the two wells after stimulation and expansion of troughs of depression. After hydraulic fracturing, the estimated total yield of the individual wells increased by 2.5 times due to the hydraulic fracturing. However, the wells may be receiving water from the same source, and well interference may affect any significant increase in their combined yield. Our analyses demonstrate the value in applying inverse analysis, derivative analysis, and diagnostic plots over the conventional method of manual type-curve analysis. In addition, our improvement in the aquifer test interpretation of the 1970’s test data has implications for more reliable estimates of sustained well yields.