Research

19 pages, 7811 KiB

Open AccessArticle

Nonlinear Tolerancing: Variation Simulation and Assembly Analysis with Regard to Contact Interaction of Parts

by Sergey Lupuleac, Margarita Petukhova, Julia Shinder, Maria Titova, Nadezhda Zaitseva and Maria Churilova

Axioms 2024, 13(1), 67; https://doi.org/10.3390/axioms13010067 - 20 Jan 2024

Viewed by 737

Abstract

The variation analysis is a key tool for ensuring the high quality assembly in the process of developing the technology for manufacturing of aircraft parts. One of the main factors in variations is the deviations in the positioning procedure. This paper is devoted [...] Read more.

The variation analysis is a key tool for ensuring the high quality assembly in the process of developing the technology for manufacturing of aircraft parts. One of the main factors in variations is the deviations in the positioning procedure. This paper is devoted to the development of an approach that allows taking into account the variations during positioning and merging it with the special algorithm of contact problem solving. The impact of varied boundary conditions is incorporated into an additional vector of forces that can be interpreted as reactions to the shift of supports. The obtained results are illustrated with a case of wing-to-fuselage assembly. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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14 pages, 583 KiB

Open AccessArticle

Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix

by Hasan Akın

Axioms 2023, 12(9), 880; https://doi.org/10.3390/axioms12090880 - 14 Sep 2023

Cited by 1 | Viewed by 1126

Abstract

In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-

(s, (2 t - 1) / 2)

with the nearest neighbors and the external magnetic field. We establish the partition function of the model [...] Read more.

In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-

(s, (2 t - 1) / 2)

with the nearest neighbors and the external magnetic field. We establish the partition function of the model using the transfer matrix. We compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the iterative equations associated with the model, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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28 pages, 3020 KiB

Open AccessArticle

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

by Heng-Pin Hsu, Jer-Rong Chang, Chih-Yuan Weng and Chun-Jung Huang

Axioms 2023, 12(7), 708; https://doi.org/10.3390/axioms12070708 - 20 Jul 2023

Viewed by 1471

Abstract

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous [...] Read more.

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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30 pages, 7747 KiB

Open AccessArticle

Fluid Dynamics Calculation in SF6 Circuit Breaker during Breaking as a Prerequisite for the Digital Twin Creation

by Vladislav V. Popovtsev, Alexandra I. Khalyasmaa and Yurii V. Patrakov

Axioms 2023, 12(7), 623; https://doi.org/10.3390/axioms12070623 - 22 Jun 2023

Cited by 3 | Viewed by 2001

Abstract

The requirements to switching the capacities of SF6 circuit breakers submitted by Russian Grid companies are difficult to satisfy. The first limitation is related to material and financial costs in order to create a new requirement-satisfying switching device. The second limitation is dictated [...] Read more.

The requirements to switching the capacities of SF6 circuit breakers submitted by Russian Grid companies are difficult to satisfy. The first limitation is related to material and financial costs in order to create a new requirement-satisfying switching device. The second limitation is dictated by the necessity of calculating complex physical processes in a circuit braker interrupter during fault–current making or breaking before creating a prototype. The latter task is reduced to the problem of simulating the processes of interaction between the switching arc and the SF6 gas flow. This paper deals with the solution of the problem both analytically by a special method and numerically by a numerical software package through the creation of a mathematical model of the interaction process. The switching arc is taken into account as a form of a temperature source, based on experimental data on measuring the temperature of the arc column. The key feature of the research is to use the finite element method based on a moving mesh—the Arbitrary Lagrangian Eulerian (ALE) method. Such a problem statement allows us to take the contact separation curve of the circuit breaker into account as the input data of the model. The calculations were carried out during fault-current breaking by a 110 kV SF6 dead-tank circuit breaker. The calculations of pressure and mass flow in the under-piston volume change, gas flow speed, and temperature depending on the contact separation are given. The proposed model of the switching arc was used to simulate the process of 25 kA symmetrical fault–current breaking and was compared with an experiment. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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26 pages, 5158 KiB

Open AccessArticle

Steam Gasification in a Fluidized Bed with Various Methods of In-Core Coal Treatment

by Nikolay Abaimov, Alexander Ryzhkov, Vladimir Tuponogov, Leonid Simbiriatin, Alexey Dubinin, Lu Ding and Sergey Alekseenko

Axioms 2023, 12(6), 587; https://doi.org/10.3390/axioms12060587 - 13 Jun 2023

Cited by 2 | Viewed by 1277

Abstract

The aim of this work is to study coal steam gasification with various methods of coal in-core treatment in FB using a newly developed thermodynamic calculation method. A calculational study of subbituminous coal steam non-catalytic gasification was carried out using four different methods [...] Read more.

The aim of this work is to study coal steam gasification with various methods of coal in-core treatment in FB using a newly developed thermodynamic calculation method. A calculational study of subbituminous coal steam non-catalytic gasification was carried out using four different methods of coal in-core treatment in single-vessel multisectional fluidized-bed gasifiers. A semi-empirical model based on the entropy maximization thermodynamic method and “restricted equilibria” based on previously obtained experimental data has been developed. Based on thermodynamic calculations, the effect of the leading thermochemical processes and operating parameters of the fluidized bed (temperature, fluidization number, steam/coal ratio feed rate) was revealed. New information was obtained regarding the composition of char and syngas at the gasifier outlet, the syngas heating value, and the cold gas efficiency of the steam gasification of Borodinskiy subbituminous coal char. The results indicate the possibility of significantly accelerating and improving non-catalytic steam gasification in fluidized bed gasifiers through the appropriate organization of in-core coal treatment. Based on the results obtained, the following recommendation is made—when designing multi-section and multi-vessel steam-blown gasifiers, the ratio of residence times should be set in favor of increasing the coal residence time in the steam-blown carbonization zone. Structurally, this can be achieved by increasing the volume and/or area of the steam-blown carbonization section (vessel). Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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12 pages, 4340 KiB

Open AccessArticle

Thermo-Optical Measurements and Simulation in a Fibre-Optic Circuit Using an Extrinsic Fabry–Pérot Interferometer under Pulsed Laser Heating

by Artem N. Kotov, Aleksandr A. Starostin, Vladimir I. Gorbatov and Pavel V. Skripov

Axioms 2023, 12(6), 568; https://doi.org/10.3390/axioms12060568 - 08 Jun 2023

Cited by 1 | Viewed by 782

Abstract

Advantages of using an external Fabry–Pérot interferometer (EFPI) as a high-speed local temperature deformation sensor are demonstrated for the fibre-optic circuit combining a powerful laser beam for surface heating with a low-power probing radiation. The difference in the formation of the heating and [...] Read more.

Advantages of using an external Fabry–Pérot interferometer (EFPI) as a high-speed local temperature deformation sensor are demonstrated for the fibre-optic circuit combining a powerful laser beam for surface heating with a low-power probing radiation. The difference in the formation of the heating and probing radiation provides a simple basis for varying the gap between the fibre end and the surface in order to change the ratio between the heating and EFPI measuring areas. Using an example of modelling the laser heating by radiation from a standard single-mode fibre, we demonstrate the possibility of employing the EFPI to measure the temperature deformation of the surface on a quasi-isothermal area with the temperature close to the maximum at gap values of more than 100 μm. With the condition of preliminary calibration, the proposed scheme can be used to evaluate the heat treatment of the surface with the speed of the applied photodetector. The practical possibilities of the method are demonstrated on examples of heating some metal and semiconductor samples by laser pulses of microsecond duration. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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16 pages, 3307 KiB

Open AccessArticle

Structural Damage Identification Using the First-Order Vibration-Mode-Based Frequency-Shift Flexibility Sensitivity Algorithm

by Shanshan Cao, Qiuwei Yang and Xi Peng

Axioms 2023, 12(6), 551; https://doi.org/10.3390/axioms12060551 - 02 Jun 2023

Cited by 1 | Viewed by 994

Abstract

The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal [...] Read more.

The diagnosis of structural damage usually belongs to a mathematical inverse problem. This work presents a novel frequency-shift flexibility sensitivity algorithm for structural damage assessment using only the first-order vibration mode to achieve the goal of successfully identifying structural damage with fewer modal parameters. The core idea of the proposed method is to make the first-order vibration mode contribute the most to a structural flexibility matrix through the frequency-shift operation. A high-precision flexibility matrix can be obtained after the frequency-shift operation, which only needs the first mode of structural free vibration. Through this special advantage, structural damage coefficients can be accurately calculated by the frequency-shift flexibility sensitivity equation. Thus, a reliable identification result can be obtained according to the values of the calculated damage coefficients. In some engineering applications, another advantage of the proposed method is that it does not require a complete finite element modeling process, as long as a few lower-frequency vibration modes of the intact structure are measured. A truss structure and a beam structure are used as two numerical examples to demonstrate the proposed approach. The results show that the proposed method has higher calculation accuracy than the ordinary flexibility sensitivity method by using only the first-order vibration mode. The proposed method can overcome possible misdiagnosis of the ordinary flexibility sensitivity method. It also has been shown that the proposed method may have the potential to identify minor damage in a structure. Using the experimental data of a steel frame structure, the effectiveness and reliability of the proposed method have been further verified. The proposed method provides a simple way for structural damage identification with only a few vibration modal data. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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15 pages, 498 KiB

Open AccessArticle

An Energy-Efficient Optimal Operation Control Strategy for High-Speed Trains via a Symmetric Alternating Direction Method of Multipliers

by Shan Ma, Feng Ma and Chaoyu Tang

Axioms 2023, 12(5), 489; https://doi.org/10.3390/axioms12050489 - 18 May 2023

Cited by 1 | Viewed by 803

Abstract

Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control [...] Read more.

Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control problem for multiple train operation is formulated to reduce the energy consumption and improve the punctuality of trains. Then, we propose an efficient algorithm based on the framework of the symmetric alternating direction method of multipliers to solve this optimization problem. Finally, numerical simulations show that the method can obtain the optimal train control sequence in fewer iterative steps compared to the alternating direction multiplier method, thus illustrating the effectiveness of the algorithm. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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21 pages, 1852 KiB

Open AccessArticle

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

by Heng-Pin Hsu, Te-Wen Tu and Jer-Rong Chang

Axioms 2023, 12(5), 416; https://doi.org/10.3390/axioms12050416 - 24 Apr 2023

Cited by 2 | Viewed by 2574

Abstract

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case [...] Read more.

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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15 pages, 11016 KiB

Open AccessArticle

A Hermite Surface Triangle Modeling Method Considering High-Precision Fitting of 3D Printing Models

by Ruichao Lian, Shikai Jing, Yang Chen and Jiangxin Fan

Axioms 2023, 12(4), 370; https://doi.org/10.3390/axioms12040370 - 11 Apr 2023

Viewed by 1441

Abstract

Three-dimensional printing is a layer-by-layer stacking process. It can realize complex models that cannot be manufactured by traditional manufacturing technology. The most common model currently used for 3D printing is the STL model. It uses planar triangles to simplify the CAD model. This [...] Read more.

Three-dimensional printing is a layer-by-layer stacking process. It can realize complex models that cannot be manufactured by traditional manufacturing technology. The most common model currently used for 3D printing is the STL model. It uses planar triangles to simplify the CAD model. This approach makes it difficult to fit complex surface shapes with high accuracy. The fitting result usually suffers from loss of local features of the model, poor fitting accuracy, or redundant data due to face piece subdivision, which will cause problems such as poor manufacturing accuracy or difficult data processing. To this end, this paper proposes a method for constructing Hermite surface models considering high-precision fitting of 3D printing models. The mapping relationship between different surface triangles and the same base triangle is established by analyzing the characteristics of Hermite surface triangles in AMF format files and using the radial variation property. By constructing a cubic surface model with general parameters and combining the vertex and tangent vector information, a cubic Hermite curve and surface triangle model are obtained. A sampling mapping point solution method is proposed, which transforms the volume integration problem between models into the summation problem of sampling point height difference. Considering the mean deviation and variance in multiple directions of the sampling points, a method for calculating and evaluating the model fitting error is constructed. Finally, the effectiveness of the proposed method is verified by rabbit and turbine. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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15 pages, 7531 KiB

Open AccessArticle

An Asymmetric Model Position Dependent Mass: Quantum Mechanical Study

by Biswanath Rath, Pravanjan Mallick, Jihad Asad, Rania Wannan, Rabab Jarrar and Hussein Shanak

Axioms 2023, 12(4), 318; https://doi.org/10.3390/axioms12040318 - 23 Mar 2023

Cited by 3 | Viewed by 1043

Abstract

We propose an asymmetric model position dependent mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the [...] Read more.

We propose an asymmetric model position dependent mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the model asymmetric mass works well for weak coupling preserving the symmetric phase portrait. However, the dominance of asymmetric feature of the mass in the system clearly visible for higher values of the constant associated with the mass. Though, both position dependent mass and potential have significant role in controlling the spectral feature of the system, one may dominate over other for certain cases. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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13 pages, 5215 KiB

Open AccessArticle

Mathematical Description of the Aerodynamic Characteristics of Stationary Flows in a Vertical Conical Diffuser When Air Is Supplied through Various Tube Configurations

by Leonid Plotnikov

Axioms 2023, 12(3), 244; https://doi.org/10.3390/axioms12030244 - 27 Feb 2023

Cited by 1 | Viewed by 925

Abstract

Conical diffusers of various configurations are used in many kinds of technical equipment and manufacturing processes. Therefore, it is a relevant objective to obtain reliable experimental and mathematical data on the aerodynamic characteristics of diffusers. This article presents experimental data on the aerodynamics [...] Read more.

Conical diffusers of various configurations are used in many kinds of technical equipment and manufacturing processes. Therefore, it is a relevant objective to obtain reliable experimental and mathematical data on the aerodynamic characteristics of diffusers. This article presents experimental data on the aerodynamics of stationary flows in a vertical conical diffuser when air is supplied through tubes with various cross sections (circle, square, and triangle). Instantaneous values of air flow velocity are measured with a constant-temperature hot-wire anemometer. Data are obtained on the velocity fields and turbulence intensity along the height and the diameter of the diffuser’s cylindrical part when air is supplied through tubes of various configurations. It is established that air supply through profiled tubes has a significant effect on the shape of the velocity field and turbulence intensity in a vertical conical diffuser. For example, higher values of turbulence intensity are typical of air supplied through profiled tubes (the differences reach 50%). A mathematical formulation (linear and exponential equations) of the change in the average speed and intensity of air flow turbulence along the height of the diffuser’s cylindrical part for various initial conditions and supply tube configurations is presented. The obtained findings will make it possible to refine mathematical models and update algorithms for engineering the design of diffusers for various engineering processes and pieces of technical equipment. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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29 pages, 17111 KiB

Open AccessArticle

LBM-MHD Data-Driven Approach to Predict Rayleigh–Bénard Convective Heat Transfer by Levenberg–Marquardt Algorithm

by Taasnim Ahmed Himika, Md Farhad Hasan, Md. Mamun Molla and Md Amirul Islam Khan

Axioms 2023, 12(2), 199; https://doi.org/10.3390/axioms12020199 - 13 Feb 2023

Cited by 7 | Viewed by 1456

Abstract

This study aims to consider lattice Boltzmann method (LBM)–magnetohydrodynamics (MHD) data to develop equations to predict the average rate of heat transfer quantitatively. The present approach considers a 2D rectangular cavity with adiabatic side walls, and the bottom wall is heated while the [...] Read more.

This study aims to consider lattice Boltzmann method (LBM)–magnetohydrodynamics (MHD) data to develop equations to predict the average rate of heat transfer quantitatively. The present approach considers a 2D rectangular cavity with adiabatic side walls, and the bottom wall is heated while the top wall is kept cold. Rayleigh–Bénard (RB) convection was considered a heat-transfer phenomenon within the cavity. The Hartmann (

H a

) number, by varying the inclination angle (

θ

), was considered in developing the equations by considering the input parameters, namely, the Rayleigh (

R a

) numbers, Darcy (

D a

) numbers, and porosity (

ϵ

) of the cavity in different segments. Each segment considers a data-driven approach to calibrate the Levenberg–Marquardt (LM) algorithm, which is highly linked with the artificial neural network (ANN) machine learning method. Separate validations have been conducted in corresponding sections to showcase the accuracy of the equations. Overall, coefficients of determination (

R^{2}

) were found to be within 0.85 to 0.99. The significant findings of this study present mathematical equations to predict the average Nusselt number (

\bar{N u}

). The equations can be used to quantitatively predict the heat transfer without directly simulating LBM. In other words, the equations can be considered validations methods for any LBM-MHD model, which considers RB convection within the range of the parameters in each equation. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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16 pages, 15923 KiB

Open AccessArticle

Aggregative Game for Distributed Charging Strategy of PEVs in a Smart Charging Station

by Ti Kang, Huaqing Li and Lifeng Zheng

Axioms 2023, 12(2), 186; https://doi.org/10.3390/axioms12020186 - 10 Feb 2023

Viewed by 1107

Abstract

This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs [...] Read more.

This paper proposes a charging strategy for plug-in electric vehicles (PEVs) in a smart charging station (SCS) that considers load constraints and time anxieties. Due to the rapidly growing load demand of PEVs and the load capacity investments in infrastructure, PEV charging needs to be subject to overload limits, beyond which failures can occur. The time anxiety is presented to address some of the uncertainties that may arise while charging PEVs. Under an aggregative game framework, this paper constructs a price-driven charging model to minimize costs by choosing the optimal charging strategy. Meanwhile, since the driver information is an aggregated item in the PEV cost function, the drivers’ privacy can be protected. Then, a distributed reflected forward–backward (RFB) splitting method is developed to search for the generalized Nash equilibria (GNE) of the game. The convergence of the proposed algorithm and the effectiveness of the charging strategy are verified by the detailed simulation and results. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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13 pages, 805 KiB

Open AccessArticle

A Fast Calculation Method for Sensitivity Analysis Using Matrix Decomposition Technique

by Qiuwei Yang and Xi Peng

Axioms 2023, 12(2), 179; https://doi.org/10.3390/axioms12020179 - 09 Feb 2023

Cited by 6 | Viewed by 1245

Abstract

The sensitivity reanalysis technique is an important tool for selecting the search direction in structural optimization design. Based on the decomposition perturbation of the flexibility matrix, a fast and exact structural displacement sensitivity reanalysis method is proposed in this work. For this purpose, [...] Read more.

The sensitivity reanalysis technique is an important tool for selecting the search direction in structural optimization design. Based on the decomposition perturbation of the flexibility matrix, a fast and exact structural displacement sensitivity reanalysis method is proposed in this work. For this purpose, the direct formulas for computing the first-order and second-order sensitivities of structural displacements are derived. The algorithm can be applied to a variety of the modifications in optimal design, including the low-rank modifications, high-rank modifications, small modifications and large modifications. Two numerical examples are given to verify the effectiveness of the proposed approach. The results show that the presented algorithm is exact and effective. Compared with the existing two reanalysis methods, this method has obvious advantages in calculation accuracy and efficiency. This new algorithm is very useful for calculating displacement sensitivity in engineering problems such as structure optimization, model correction and defect detection. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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