Proposed Sensitivity Analysis Method Research Articles

EMsFEM based concurrent topology optimization method for hierarchical structure with multiple substructures

An efficient concurrent topology optimization method based on the Extended Multiscale Finite Element Method (EMsFEM) is proposed to design the hierarchical structure with multiple substructures in this paper. Firstly, the mathematical description model for the topology of hierarchical structures and the interpolation model for material are established at both the macro and substructural levels. Then the concurrent topology optimization formulation for hierarchical structures is proposed based on EMsFEM. After that, the sensitivity analysis scheme of the objective function and constraint functions is given, based on which the Method of Moving Asymptotes (MMA) is employed to update design variables. The assumption of separation of length scales is not required in the proposed method, which avoids the disconnection between adjacent substructures. Besides, without limiting the topology and spatial distribution of substructures in advance, the concurrent topology optimization for hierarchical structures with multiple substructures can be realized using this method. The provided numerical examples verify the correctness and effectiveness of the proposed sensitivity analysis method as well as the feasibility, effectiveness and scalability of the proposed concurrent topology optimization method for hierarchical structures. Finally, the influence of the initial substructure configurations and the number of substructure types on the final configuration and mechanical performance of hierarchical structures are explored with a series of numerical examples.

A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head

Geometric errors are the main factors affecting the output accuracy of the parallel spindle head, and it is necessary to perform a sensitivity analysis to extract the critical geometric errors. The traditional sensitivity analysis method analyzes the output position and orientation errors independently, defining multiple sensitivity indices and making it difficult to determine critical geometric errors. In this paper, we propose sensitivity indices that can comprehensively consider position and orientation errors. First, the configuration of the hybrid machine tool is introduced, and the TCP position error model is derived. Then, the tool radius and the effective cutting length are introduced, and the sensitivity indices are defined. After that, the sensitivity analysis of the 3-DOF parallel spindle head is performed using the proposed sensitivity indices, and six critical geometric errors are extracted. The machining accuracy of the parallel spindle head can be greatly improved by improving the critical geometric errors. The proposed sensitivity analysis method can provide important guidance for machine tool accuracy design.

Recursive inverse dynamics sensitivity analysis of open-tree-type multibody systems

We present a first-order recursive approach to sensitivity analysis based on the application of the direct differentiation method to the inverse Lagrangian dynamics of rigid multibody systems. Our method is simple and efficient and is characterized by the following features. Firstly, it describes the kinematics of multibody systems using branch connectivity graphs and joint-branch connectivity matrices. For most mechanical systems with an open-tree kinematic structure, this method turns out to be more efficient compared to other kinematic descriptions employing joint or link connectivity graphs. Secondly, a recursive sensitivity analysis is presented for a dynamic system with an open-tree kinematic structure and inverse dynamic equations described in terms of the Lagrangian formalism. Thirdly, known approaches to recursive inverse dynamic and sensitivity analyses are modified to include dynamic systems with external forces and torques acting simultaneously at all joints. Finally, the proposed method for sensitivity analysis is easy to implement and computationally efficient. It can be utilized to evaluate the derivatives of the dynamic equations of multibody systems in gradient-based optimization algorithms. It also allows less experienced users to perform sensitivity analyses using the power of high-level programming languages such as MATLAB. To illustrate the method, simulation results for a human body model are discussed. The shortcomings of the method and possible directions for future work are outlined.

Sensitivity-Based Topology Optimization of Squirrel-Cage Induction Motor in Time Domain Using Multi-Material Level-Set Method

Although the output power of an induction motor (IM) is generally lower than that of the interior permanent magnet synchronous motor, IMs are widely used in industrial world due to their solidity, low cost, self-starting, etc. For designing high-performance IMs, a computer-aided engineering approach using the finite-element method is essential. However, because a finite-element analysis of IMs takes a long time due to the long transient state in the time domain, it is difficult to perform the optimization using evolutionary algorithms. To overcome this difficulty, a sensitivity-based topology optimization (TO) method for IMs in the frequency domain was proposed. However, this method has a drawback: because the magnetic nonlinearity in the rotor and stator cores cannot be considered, the physical quantities also cannot be accurately estimated. To consider the magnetic nonlinearity in the TO of the IM, the sensitivity analysis with the rotor rotation of the IM using the time-domain adjoint variable method is proposed here. The proposed sensitivity analysis method for the steady state of the IM was evaluated, and it was applied to the TO of a squirrel-cage IM. The effective rotor structure, which is composed of the iron core and secondary conductor, was derived from the multi-material level-set method.

Uncertainty Analysis and Sensitivity Estimation on an Artillery External Ballistic System

Abstract In the design of artillery external ballistics, sensitivity analysis can effectively quantify the influence of multi-source uncertain parameters on the dispersion of projectile landing points to improve the precise attack ability of artillery. However, for a complicated artillery external ballistic system containing multiple inputs and outputs, its mapping relationships are not definite under uncertainty and it is difficult to estimate a comprehensive sensitivity index due to involving the calculation of high dimensional integral. Therefore, a sensitivity analysis method based on the combination of variance and covariance decomposition with the approximate high dimensional model representation (AHDMR) is proposed to measure the influence of muzzle state parameters, projectile characteristic parameters, etc. on projectile landing points under uncertainty in this paper. First, we establish the numerical simulation model of artillery external ballistics by combing the external ballistic theory and Runge–Kutta algorithm to acquire the mapping relationships between the uncertain input parameters and the dispersion of projectile landing points and implement uncertainty analysis under different uncertainty levels (UL) and distributions. Then, with the use of a set of orthogonal polynomials for uniform and Gaussian distribution, respectively, the high dimensional model representation of the mapping relationship is approximately expressed and the compressive sensitivity indices can be effectively estimated based on the Monte Carlo simulation. Moreover, the comparison results of two numerical examples indicate the proposed sensitivity analysis method is accurate and practical. Finally, through the method, the importance rankings of multi-uncertain parameters on projectile landing points for two distributions are effectively quantified under the UL = [0.01, 0.02, 0.03, 0.04, 0.05].

Towards an Explainable Model for Sepsis Detection Based on Sensitivity Analysis

ObjectivesSepsis is a life-threatening condition which is responsible for a high proportion of intra-hospital deaths and related healthcare costs each year. Early detection and treatment of sepsis episodes is critical, since an early treatment may highly improve prognosis. This study proposed an original method to increase the interpretability of a set of machine learning models for the early detection of sepsis onset. Material and methodsOpen data from the electronic medical records of 40,336 patients monitored in intensive care units (ICU), provided by the PhysioNet/Computing in Cardiology Challenge 2019 is used in this paper. We proposed a method including data preprocessing, feature engineering, model construction and tuning, as well as an original interpretability analysis method for the final stage. ResultsA total of 24 models were developed and analyzed. The best model, based on 142 features achieved a 0.4274 utility score. The best compact model integrates only 20 selected features, and provided a utility score of 0.3862. Meanwhile, the proposed sensitivity analysis method allows for the identification of the most relevant markers to early detect the onset of sepsis, as well as their interdependence and relative importance on the final decision. ConclusionA set of optimized machine-learning models were proposed for predicting sepsis early in a real-time way with high performance, and interpretable information including the most significant biomarkers were analyzed through novel interpretability method.

Towards rigorous boundary value level sensitivity analyses using FEM

The increased complexity of contemporary constitutive models for soils requires a rigorous method to evaluate the effect of the large number of model parameters on the results. Ideally, the interaction effects between the individual parameters should also be quantified. This is achieved by combining a state-of-the-art global sensitivity method with a general purpose Finite Element Method (FEM) for geotechnics. The method is tested for the non-trivial example of coupled hydro-mechanical response of clay in oedometric compression. The results indicate that proposed method for rigorous sensitivity analysis provides a feasible, yet more powerful, alternative to the method commonly used by engineers, i.e. the sequences of one-factor-at-a-time (OFAT) trails.

Transition prediction and sensitivity analysis for a natural laminar flow wing glove flight experiment

Natural laminar flow technology can significantly reduce aircraft aerodynamic drag and has excellent technical appeal for transport aircraft development with high aerodynamic efficiency. Accurately and efficiently predicting the laminar-to-turbulent transition and revealing the maintenance mechanism of laminar flow in a transport aircraft’s flight environment are significant for developing natural laminar flow wings. In this research, we carry out natural laminar flow flight experiments with different Reynolds numbers and angles of attack. The critical N-factor is calibrated as 9.0 using flight experimental data and linear stability theory from a statistical perspective, which makes sure that the relative error of transition location is within 5%. We then implement a simplified eN transition prediction method with a similar accuracy compared with linear stability theory. We compute the sensitivity information for the simplified eN method with an adjoint-based method, using the automatic differentiation technique (ADjoint). The impact of Reynolds numbers and pressure distributions on TS waves is analyzed using the sensitivity information. Through the sensitivity analysis, we find that: favorable pressure gradients not only suppress the development of TS waves but also decrease their sensitivity to Reynolds numbers; there exist three special regions which are very sensitive to the pressure distribution, and the sensitivity decreases as the local favorable pressure gradient increases. The proposed sensitivity analysis method enables robust natural laminar flow wings design.

A Feasible Delay Margin Sensitivity Analysis Method

This letter proposes a method for calculating the sensitivity of delay margin with respect to system parameters. According to the property that the delay margin is achieved when the rightmost eigenvalues of the close-loop system locate exactly on the imaginary axis, a characteristic equation with inclusion of the delay margin is established. By differentiating the characteristic equation and further introducing normalization condition for the eigenvector, the sensitivity of the delay margin with respect to any given parameters can be obtained by solving a set of sparse linear equations. Due to the general differential and algebraic equation used in the derivation, the proposed delay margin sensitivity analysis method applies to various power system models and control problems.

Geometric error sensitivity analysis for a 6-axis welding equipment based on Lie theory

The influence of geometric errors on the accuracy of machine tools tends to attract more attention to the increasing demand for high-precision machining. In this paper, geometric error modeling and sensitivity analysis are employed to quantify the importance of geometric error for a new efficient and automatic 6-axis welding equipment. The geometric error model of the 6-axis welding equipment with 36 geometric error components is established based on Lie theory. Based on the geometric error model, the new sensitivity analysis method, in which the deviation of the welding torch pose is treated as a distance metric in SE(3), is proposed to evaluate the influences of geometric errors on the accuracy of the welding torch. And the sensitivity coefficient of each geometric error is derived by considering the basic value of geometric errors. Numerical simulations of a typical welding trajectory for intersecting pipes are conducted to analyze the sensitivity of geometric errors by the new method. The simulation results verified the validity of the sensitivity analysis method and the dominant geometric errors affecting the accuracy of the welding equipment were identified. Compared with the previous sensitivity analysis method, the proposed sensitivity analysis method considers the orientation error and position error of the welding torch simultaneously, which is more convenient and effective, and can also be applied in precision design and geometric error compensation of machine tools.

Data-driven sensitivity analysis of complex machine learning models: A case study of directional drilling

Classical sensitivity analysis of machine learning regression models is a topic sparse in literature. Most of data-driven models are complex black boxes with limited potential of extracting mathematical understanding of underlying model self-arranged through the training algorithm. Sensitivity analysis can uncover erratic behavior stemming from overfitting or insufficient size of the training dataset. It can also guide model evaluation and application. In this paper, our work on data-driven sensitivity analysis of complex machine learning models is presented. Rooted in one-at-a-time method it utilizes training, validation and testing datasets to cover the hyperspace of potential inputs. The method is highly scalable, it allows for sensitivity analysis of individual as well as groups of inputs. The method is not computationally expensive, scaling linearly both with the available data samples, and in relation to the quantity of inputs and outputs. Coupled with the fact that calculations are considered embarrassingly parallel, it makes the method attractive for big models. In the case study, a regression model to predict inclinations using recurrent neural network was employed to illustrate our proposed sensitivity analysis method and results.

Sensitivity analysis of relationship between error motions and machined shape errors in five-axis machining center - Peripheral milling using square-end mill as test case -

Five-axis machine tools, consisting of three translational axes and two rotary axes, are increasingly being used for complex surface machining. This paper proposes a new sensitivity analysis method to elucidate the relationship between the tool trajectory error and the error motions of the feed axes. Based on the free-curve trajectory during simultaneous five-axis machining, a surface coordinate system is created for each tool center point, to define the tool trajectory and the trajectory errors. Then, a novel sensitivity coefficient is defined to investigate the relationship between the trajectory error and the error motions. It is shown that the proposed sensitivity analysis method can successfully determine whether the trajectory is sensitive to the error motions, based on sensitivity analyses performed during conic frustum machining and S-shaped machining tests. Moreover, the sensitivity analysis method can also predict the effects of the error motion source, such as the reversal errors. In the future, we intend to study other types of machining processes, such as ball-end milling, as well.

Strategic Framework for Parameterization of Cell Culture Models

Global Sensitivity Analysis (GSA) is a technique that numerically evaluates the significance of model parameters with the aim of reducing the number of parameters that need to be estimated accurately from experimental data. In the work presented herein, we explore different methods and criteria in the sensitivity analysis of a recently developed mathematical model to describe Chinese hamster ovary (CHO) cell metabolism in order to establish a strategic, transferable framework for parameterizing mechanistic cell culture models. For that reason, several types of GSA employing different sampling methods (Sobol’, Pseudo-random and Scrambled-Sobol’), parameter deviations (10%, 30% and 50%) and sensitivity index significance thresholds (0.05, 0.1 and 0.2) were examined. The results were evaluated according to the goodness of fit between the simulation results and experimental data from fed-batch CHO cell cultures. Then, the predictive capability of the model was tested against four different feeding experiments. Parameter value deviation levels proved not to have a significant effect on the results of the sensitivity analysis, while the Sobol’ and Scrambled-Sobol’ sampling methods and a 0.1 significance threshold were found to be the optimum settings. The resulting framework was finally used to calibrate the model for another CHO cell line, resulting in a good overall fit. The results of this work set the basis for the use of a single mechanistic metabolic model that can be easily adapted through the proposed sensitivity analysis method to the behavior of different cell lines and therefore minimize the experimental cost of model development.

A sensitivity method to analyze the volumetric error of five-axis machine tool

Geometric errors have a comprehensive influence on the volumetric error of the five-axis machine tool. The identification of the vital geometric errors that have major effect on the volumetric error is the key problem to improve the machine tool accuracy. Therefore, a sensitivity analysis method to identify the vital geometric errors is proposed in this paper. The volumetric error model of a five-axis machine tool with 41 geometric errors is established based on the multi-body system method. Through the projection of the error vectors and the introduction of the effective cutting length, LSIL and GSIL are defined as the sensitivity indices for the local and global sensitivity analysis, respectively. Simulations are conducted for the local and global sensitivity analysis in which the LSIL and GSIL are used. And the analysis results have proven the validity of the proposed sensitivity analysis method. Compared with the conventional sensitivity analysis method, the proposed sensitivity analysis method has considered the position and posture error of the cutting tool simultaneously, which is more effective and efficient. The analysis results are helpful to improve the accuracy of the five-axis machine tool.

Global sensitivity analysis via multi-fidelity polynomial chaos expansion

The presence of uncertainties is inevitable in engineering design and analysis, where failure in understanding their effects might lead to the structural or functional failure of the systems. The role of global sensitivity analysis in this aspect is to quantify and rank the effects of input random variables and their combinations to the variance of the random output. In problems where the use of expensive computer simulations is required, metamodels are widely used to speed up the process of global sensitivity analysis. In this paper, a multi-fidelity framework for global sensitivity analysis using polynomial chaos expansion (PCE) is presented. The goal is to accelerate the computation of Sobol sensitivity indices when the deterministic simulation is expensive and simulations with multiple levels of fidelity are available. This is especially useful in cases where a partial differential equation solver computer code is utilized to solve engineering problems. The multi-fidelity PCE is constructed by combining the low-fidelity and correction PCE. Following this step, the Sobol indices are computed using this combined PCE. The PCE coefficients for both low-fidelity and correction PCE are computed with spectral projection technique and sparse grid integration. In order to demonstrate the capability of the proposed method for sensitivity analysis, several simulations are conducted. On the aerodynamic example, the multi-fidelity approach is able to obtain an accurate value of Sobol indices with 36.66% computational cost compared to the standard single-fidelity PCE for a nearly similar accuracy.

Orbit uncertainty propagation and sensitivity analysis with separated representations

Most approximations for stochastic differential equations with high-dimensional, non-Gaussian inputs suffer from a rapid (e.g., exponential) increase of computational cost, an issue known as the curse of dimensionality. In astrodynamics, this results in reduced accuracy when propagating an orbit-state probability density function. This paper considers the application of separated representations for orbit uncertainty propagation, where future states are expanded into a sum of products of univariate functions of initial states and other uncertain parameters. An accurate generation of separated representation requires a number of state samples that is linear in the dimension of input uncertainties. The computation cost of a separated representation scales linearly with respect to the sample count, thereby improving tractability when compared to methods that suffer from the curse of dimensionality. In addition to detailed discussions on their construction and use in sensitivity analysis, this paper presents results for three test cases of an Earth orbiting satellite. The first two cases demonstrate that approximation via separated representations produces a tractable solution for propagating the Cartesian orbit-state uncertainty with up to 20 uncertain inputs. The third case, which instead uses Equinoctial elements, reexamines a scenario presented in the literature and employs the proposed method for sensitivity analysis to more thoroughly characterize the relative effects of uncertain inputs on the propagated state.

Geometric error modeling and sensitivity analysis of a five-axis machine tool

It is widely acknowledged that precision design and error compensation are two basic methods to achieve high geometrical accuracy for machine tools. And for these two methods, error modeling and sensitivity analysis are key issues. In this paper, an error modeling method based on multi-body system (MBS) is used to construct the error mapping between error sources and the cutting tool pose error for a five-axis machine tool. According to the error mapping, the traditional definitions of local sensitivity indices (LSIs) and global sensitivity indices (GSIs) are introduced. Based on the LSIs and GSIs, the general local sensitivity indices (GLSIs), general global sensitivity indices (GGSIs), and general global sensitivity fluctuation indices (GGSFIs) are proposed. By using these new indices, error sensitivity analysis of the five-axis machine tool is conducted. According to the error sensitivity analysis results, the precision design of angular error components is conducted in numerical simulation. The results show that by using the proposed sensitivity analysis method, we can improve less error components but get more improvement for the cutting tool accuracy. It indicates that the proposed sensitivity indices and sensitivity analysis method are very effective and meaningful. The proposed sensitivity indices and sensitivity analysis method can also be used in the precision design and error compensation for other machine tools.

Sensitivity analysis of a seismic risk scenario using sparse Chebyshev polynomial expansion

S U M M A R Y Global parametric sensitivity analysis of a complete seismic risk scenario is performed by means of the response surface method based on sparse Chebyshev polynomial expansion. Hazard model, described via a mixed technique combining Green’s function and stochastic methods, is used to simulate free-field seismic records. Vulnerability model, based on the period-dependent damage spectra, is constructed using principles of normalized hysteretic energy and displacement ductility. Approximation of the risk model by means of orthogonal polynomials allows to compute sensitivity indices for model parameters directly from the interpolation matrix. The Monte Carlo method runs on top of the meta-model and allows to build probability density functions of the damage index at target period. The proposed method for sensitivity analysis is useful for understanding the impact of aleatory uncertainties on the risk model outputs.

Stochastic Sensitivity Analysis for Models with Parameter Correlation and Model Approximation Method

In the stochastic sensitivity analysis, a large number of simulation models lead to low computational e‐ciency. With the dimension increasing in assigned problems, the accurate is di‐cult to achieve by popular regression methodologies. Without considering the correlations of parameters, inaccurate sensitivity coe‐cients would be calculated by analyzing the efiect of parametric variables on structures, moreover, the existing methods only calculate the local gradient as sensitivity. According to these problems, approximate model and global sensitivity method are employed for design sensitivity analysis of structures. The approximate model is constructed by the hybrid neural network which possesses signiflcant learning capacity and generalization capability with a small amount of information. The improved chaotic particle swarm is applied to optimize the neural network so as to improve the computational e‐ciency and accuracy. The proposed sensitivity analysis method values the global response of the outputs by varying all the input parameters at a time with the correlations of parameters. Uniform design and Latin hypercube sampling are used to sample points. Numerical analysis show that the proposed method can successfully measure the actual sensitivity.

Design of Star-Shaped Flextensional Stator for Ultrasonic Motor

When a driving voltage opposite to the piezoelectric polarity is applied on the flextensional stator, it will generate the normal force, of which the operating voltage range of piezoelectric actuators will decrease. This paper presents a novel stator design for producing the normal force in which the driving voltage has the same piezoelectric polarity, which is based on the structure of two multilayer piezoelectric actuators clamped in a star-shaped shell. To obtain the two close resonance frequencies of flexural and translation modes, a genetic algorithm combined with the finite element analysis is employed to find the optimal dimensions for the geometry of the stator. The importance of each design parameter is evaluated through a proposed sensitivity analysis method. A prototype resulting from the optimal design was fabricated and the experimental results are given to show that the stator can generate, in practice, the required coupling resonance mode between 35.15 kHz and 36.49 kHz.