Finite Element Model Reduction Research Articles

Enhancing structural analysis efficiency: a comprehensive review and experimental validation of advanced submodeling techniques, introducing the submodeling-density-shape-element removal (S-D-S-ER) method

PurposeThe paper's goal is to examine and illustrate the useful uses of submodeling in finite element modeling for topology optimization and stress analysis. The goal of the study is to demonstrate how submodeling – more especially, a 1D approach – can reliably and effectively produce ideal solutions for challenging structural issues. The paper aims to demonstrate the usefulness of submodeling in obtaining converged solutions for stress analysis and optimized geometry for improved fatigue life by studying a cantilever beam case and using beam formulations. In order to guarantee the precision and dependability of the optimization process, the developed approach will also be validated through experimental testing, such as 3-point bending tests and 3D printing. Using 3D finite element models, the 1D submodeling approach is further validated in the final step, showing a strong correlation with experimental data for deflection calculations.Design/methodology/approachThe authors conducted a literature review to understand the existing research on submodeling and its practical applications in finite element modeling. They selected a cantilever beam case as a test subject to demonstrate stress analysis and topology optimization through submodeling. They developed a 1D submodeling approach to streamline the optimization process and ensure result validity. The authors utilized beam formulations to optimize and validate the outcomes of the submodeling approach. They 3D-printed the optimized models and subjected them to a 3-point bending test to confirm the accuracy of the developed approach. They employed 3D finite element models for submodeling to validate the 1D approach, focusing on specific finite elements for deflection calculations and analyzed the results to demonstrate a strong correlation between the theoretical models and experimental data, showcasing the effectiveness of the submodeling methodology in achieving optimal solutions efficiently and accurately.FindingsThe findings of the paper are as follows: 1. The use of submodeling, specifically a 1D submodeling approach, proved to be effective in achieving optimal solutions more efficiently and accurately in finite element modeling. 2. The study conducted on a cantilever beam case demonstrated successful stress analysis and topology optimization through submodeling, resulting in optimized geometry for enhanced fatigue life. 3. Beam formulations were utilized to optimize and validate the outcomes of the submodeling approach, leading to the successful 3D printing and testing of the optimized models through a 3-point bending test. 4. Experimental results confirmed the accuracy and validity of the developed submodeling approach in streamlining the optimization process. 5. The use of 3D finite element models for submodeling further validated the 1D approach, with specific finite elements showing a strong correlation with experimental data in deflection calculations. Overall, the findings highlight the effectiveness of submodeling techniques in achieving optimal solutions and validating results in finite element modeling, stress analysis and optimization processes.Originality/valueThe originality and value of the paper lie in its innovative approach to utilizing submodeling techniques in finite element modeling for structural analysis and optimization. By focusing on the reduction of finite element models and the creation of smaller, more manageable models through submodeling, the paper offers designers a more efficient and accurate way to achieve optimal solutions for complex problems. The study's use of a cantilever beam case to demonstrate stress analysis and topology optimization showcases the practical applications of submodeling in real-world scenarios. The development of a 1D submodeling approach, along with the utilization of beam formulations and 3D printing for experimental validation, adds a novel dimension to the research. Furthermore, the paper's integration of 1D and 3D submodeling techniques for deflection calculations and validation highlights the thoroughness and rigor of the study. The strong correlation between the finite element models and experimental data underscores the reliability and accuracy of the developed approach. Overall, the originality and value of this paper lie in its comprehensive exploration of submodeling techniques, its practical applications in structural analysis and optimization and its successful validation through experimental testing.

Reduction of the finite element model of a gas turbine engine rotor using CMS technique

This paper is devoted to the problems of modeling the vibration behavior of rotors of gas turbine engines. The necessity of modeling their dynamic behavior is caused by their strong influence on the strength characteristics of the engine and its resource. The complexity of solving this problem lies in the fact that the design of modern gas turbine engines has many features that do not allow the use of only one-dimensional models, and the application of three-dimensional models requires a lot of computing resources. Thus, it is suggested to use methods of reduction of finite element models for solving the problems in three-dimensional statement. This paper uses the mode component synthesis (CMS) method to reduce the dimensionality of the rotor model. It also compares the speed of solving modal analysis tasks and determining critical frequencies using complete and reduced models. The quality of reduction is determined. Conclusions about the efficiency of the reduction technique aimed at solving rotor dynamics problems are made.

Three-dimensional magnetotelluric modeling using the finite element model reduction algorithm

This study presents a fast and efficient vector finite element method, based on the model reduction algorithm, to simulate the 3-D magnetotelluric response. Firstly, we derive a finite element governing equation based on electric field vector from Maxewell's equations Then we express the electric field by a function of frequency parameter to get a transfer function that can be approximated by the Krylov subspace model reduction algorithm. The transfer function after model reduction has far lower dimensions than the original one, and it can be directly solved by the Pardiso solver. Therefore, the computational efficiency is significantly improved. For validation, we compare the proposed algorithm with the 3-D direct finite element method (3DFEM) by a three-layer model, and the results show that the computation time of the former is less than 1/10 of the latter for the same accuracy. We also compare the electromagnetic response obtained by the proposed method with that obtained by two classical models, COMMEMI 3D-2A Model and Dublin Test Model 1. These results agree well, indicating that the proposed method is feasible and efficient for 3-D magnetotelluric forward modeling.

Condensation modeling of the linear structure with nonlinear boundary conditions

The problem of modeling for the linear structure with nonlinear boundary conditions in a more efficient manner is considered by proposing a novel condensation modeling method. The key idea of the proposed method is to reduce the overall number of degrees-of-freedom (DOFs) for the nonlinear jointed structure being analyzed, which can be implemented in the following two steps. Firstly, linear DOFs of the structure are truncated via finite element model reduction by using the free-interface mode synthesis method, as numerical integration of governing equations of structures with large numbers of DOFs is always time-consuming. Secondly, nonlinear DOFs at the boundary interface are reduced via coordinate condensation, so as to further minimize the model size of the structure being analyzed and make the subsequent analysis computationally efficient and memory economical. The performance of the proposed condensation modeling method is validated via case studies focused on an experimental structure with adjustable boundary conditions. Comparative results demonstrate that the computational efficiency can be extremely improved while the accuracy is simultaneously guaranteed by using the proposed method to model the linear structure with nonlinear boundary conditions.

Finite Element Model Reduction Applied to Nonlinear Impact Simulation for Squeak and Rattle Prediction

<div class="section abstract"><div class="htmlview paragraph">Increasing demand for simulation accuracy often leads to increased finite element model complexity, which in turn, results in higher computational costs. As a provision, component mode synthesis approaches are employed to approximate the system response by using dynamic substructuring and model reduction techniques in linear systems. However, the use of available model reduction techniques in nonlinear problems containing the contact type of nonlinearities remains an interesting topic. In this paper, the application of a component mode synthesis method in squeak and rattle nonlinear simulation has been investigated. Critical regions for squeak and rattle of the side door model of a passenger car were modelled by nonlinear contact definition in finite element simulation. Craig-Bampton model reduction method was employed to substructure the finite element model while keeping the nonlinear contacts in the model. The model response was evaluated using the modal assurance criterion, frequency response analysis and contact force magnitude in comparison with the baseline model. Results showed that a great reduction in computational time (about 98%) can be achieved while the accuracy of the system response was maintained at an acceptable range for the intended application for squeak and rattle simulation. Although the prediction of impact events in time was done accurately, the contact force magnitude was estimated with average error of 2.5% to 22%, compared with the baseline results. The outcomes of the study show that to empower squeak and rattle prediction by including contact interfaces in finite element simulations, implementation of the model reduction approach can compensate the simulation cost.</div></div>

Hybrid Modeling of Nonlinear-Jointed Structures via Finite-Element Model Reduction and Deep Learning Techniques

In engineering practice many structures are assembled by several linear components through nonlinear joints. A novel hybrid modeling method based on finite element model reduction and deep learning techniques is proposed to meet the ever-increasing requirements of efficient and accurate modeling for nonlinear jointed structures. The main idea of the hybrid modeling method for nonlinear jointed structures is summarized as follows: Firstly, finite element models of linear components are reduced to improve the computing efficiency using the free-interface mode synthesis method, as numerical integration of governing equations of nonlinear structures with large numbers of degrees-of-freedom is always time-consuming. Secondly, deep neural networks are used to equivalently represent the nonlinear joints which are difficult to describe by accurate and physically-motivated models, so as to avoid the errors caused by traditional mechanism modeling or system identification. Nonlinear joints are finally replaced with their equivalent neural networks and connected with the substructure models of linear components through the compatibility of displacements and equilibrium of forces at the interfaces. The performance of the proposed hybrid modeling method is tested and assessed via a case study focused on a cantilever plate with nonlinear joints. Comparative results demonstrate the capability of the proposed method for efficient and accurate modeling of nonlinear jointed structures and predicting their intrinsic nonlinear behavior.

A time-domain finite element model reduction method for viscoelastic linear and nonlinear systems

MANY AUTHORS HAVE SHOWN THAT THE EFFECTIVE DESIGN OF VISCOELASTIC SYSTEMS CAN BE CONVENIENTLY CARRIED OUT BY USING MODERN MATHEMATICAL MODELS TO REPRESENT THE FREQUENCY- AND TEMPERATURE-DEPENDENT BEHAVIOR OF VISCOELASTIC MATERIALS. HOWEVER, IN THE QUEST FOR DESIGN PROCEDURES OF REAL-WORD ENGINEERING STRUCTURES, THE LARGE NUMBER OF EXACT EVALUATIONS OF THE DYNAMIC RESPONSES DURING ITERATIVE PROCEDURES, COMBINED WITH THE TYPICALLY HIGH DIMENSIONS OF LARGE FINITE ELEMENT MODELS, MAKES THE NUMERICAL ANALYSIS VERY COSTLY, SOMETIMES UNFEASIBLE. IT IS ESPECIALLY TRUE WHEN THE VISCOELASTIC MATERIALS ARE USED TO REDUCE VIBRATIONS OF NONLINEAR SYSTEMS. AS A MATTER OF FACT, WHICH THE RESOLUTION OF THE RESULTING NONLINEAR EQUATIONS OF MOTION WITH FREQUENCY- AND TEMPERATURE-DEPENDENT VISCOELASTIC DAMPING FORCES IS AN INTERESTING, BUT HARD-TO-SOLVE PROBLEM. THOSE DIFFICULTIES MOTIVATE THE PRESENT STUDY, IN WHICH A TIME-DOMAIN CONDENSATION STRATEGY OF VISCOELASTIC SYSTEMS IS ADDRESSED, WHERE THE VISCOELASTIC BEHAVIOR IS MODELED BY USING A FOUR PARAMETER FRACTIONAL DERIVATIVE MODEL. AFTER THE DISCUSSION OF VARIOUS THEORETICAL ASPECTS, THE EXACT AND REDUCED TIME RESPONSES ARE CALCULATED FOR A THREE-LAYER SANDWICH PLATE BY CONSIDERING NONLINEAR BOUNDARY CONDITIONS.

An inverse analysis approach based on a POD direct model for the mechanical characterization of metallic materials

Indentation tests are frequently employed to determine the mechanical properties of materials and are particularly suitable when dealing with small components due to their “quasi-non-destructiveness”. During an indentation test, the material is subjected to a triaxial stress state and, because of this, the mechanical properties cannot be inferred directly from the experiments. In these cases, suitable procedures must be implemented to derive them and, among the different alternatives available in the literature, the inverse analysis approach has been extensively studied and applied to a wide range of materials and systems. When the inverse analysis procedure relies on the use of a time-consuming finite element model for the modelling of an experimental test, the results are accurate yet computationally demanding. In the present paper, a numerically efficient approach was implemented, which relied on the use of an ‘a priori’ finite element model reduction procedure. This consists of a Proper Orthogonal Decomposition (POD) model that simulates the response of the material when subjected to an indentation test. The experimental data used in this study as input to the inverse analysis approach consist of both indentation curves and pile-up markings observed at the end of the indentation in the samples. Independent tensile tests were carried out on samples of aluminium alloys AA 6061-O and AA 7075-O. These experimental results were used for the validation of the proposed numerical approach. These results highlighted the accuracy and efficiency of the proposed procedure.

New structural damage-identification method using modal updating and model reduction

Early prediction of damages using vibration signal is essential in avoiding the failure in structures. Among different damage-detection approaches, the finite-element model updating and modal analysis-based methods are of most importance due to their applicability and feasibility. Owing to some restrictions in nodal measurements in experimental cases, finite-element model reduction is an indispensable part of fault-detection methods. Even though model reduction of dynamic systems leads to the less complicated models, an improved convergence rate and acceptable accuracy are highly required for a successful structural health monitoring of the real complex systems. In this paper, the aim is to design a damage-detection algorithm based on a new model updating method, which has a faster rate of convergence and higher accuracy. Then the proposed method is applied on a simulated damaged beam considering different noise levels to see how capable the method is in dealing with noise-corrupted data. Finally, the experimentally extracted data from a cracked beam in a real noisy condition are used to evaluate the efficiency of the proposed method in identifying the damages in a beam-like structure. It is concluded that the identification of the damages by the proposed method is encouraging and robust to the noise compared with the traditional method. Also, the proposed method converges faster and is more accurate in identifying damage than the traditional method.

Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring

Model reduction techniques such as the proper orthogonal decomposition (POD) method can be used to reduce the computational effort of simulations with a very large number of degrees-of-freedom. The reduction of finite element models including elastoplastic material behavior is still far from being trivial and inevitably leads to problems of efficiency and accuracy. One aim of the present paper is to combine the two methods--sub-structuring and POD-based model reduction--to overcome these difficulties. A typical field of application where plasticity dominates the material behavior are forming processes. The second aim of the paper is to investigate the applicability of the new approach in this context. The presented combined approach selective POD (SPOD) where the reduction is only applied in sub-domains with approximately elastic behavior shows higher accuracy than the general POD method. In this paper, the SPOD method is extended by an adaptive method of sub-structuring (A-SPOD) in which the sub-domain where model reduction is applied is determined automatically. While achieving errors of the same magnitude as by means of SPOD, the computational effort can be reduced significantly by using the A-SPOD approach.

Finite Element Model Reduction and Model Updating of structures for Control

Estimation and control of unmeasurable performance variables in complex large-scale systems is an important issue in systems and control. The preferred solution to this problem is to have a relatively low-order and accurate standard plant model which can be used for control purposes. For this purpose, a two-step procedure is proposed. The first step is to generate a reduced Finite Element (FE) model based on the selection of desired Degrees Of Freedom (DOFs), resulting in reduced-order mass, damping, and stiffness matrices. The second step is updating of the reduced-order FE model that is carried out to minimize the differences between the model and the measurements from the structure with the focus on input-output behavior. The presented approach helps to create sufficiently accurate reduced-order dynamic models which can be used for control purposes. The approach will be examined on a planar plate FE model.

Improved simulation of bearing faults using a finite element model reduction technique

Improved simulation of bearing faults using a finite element model reduction technique

Gearbox bearing fault simulation using a finite element model reduction technique

The dynamics of a mechanical system such as a gearbox assembly comprising shafts, gears and bearings can be simulated using Lumped Parameter Models (LPMs). Finite Element Method (FEM) reduction techniques based on the Craig-Bampton method of Component Mode Synthesis (CMS) are useful in creating more accurate dynamic models. These models, despite having more degrees-of-freedom for the individual components than the LPM, make very much larger FE models computationally tractable. In this paper both these approaches, namely LPM and reduced FEM, are compared to create a dynamic model of a gearbox. Earlier simulation models (both LPM and combined LPM and reduced FEM) are further improved to better match the geometry of the bearing faults used in the experimental measurements, and the experimental results from a gearbox test rig. The dynamic model is used to simulate the vibration signals in the presence of localised inner and outer race faults. The new results show better correspondence with the measured signals, in particular with respect to the detailed response to entry and exit from the fault, which can be used to determine fault size. The paper highlights the plausibility of fault simulation in Machine Condition Monitoring (MCM) where a large amount of data can be gathered without experiencing large numbers of actual failures or carrying out costly and time consuming experiments until failure with seeded faults. The simulation data can be used to train neural networks to automate the diagnostic and prognostic processes.

Reduced-Order Models of Finite Element Approximations of Electromagnetic Devices Exhibiting Statistical Variability

A methodology is proposed for the development of reduced-order models of finite element approximations of electromagnetic devices exhibiting uncertainty or statistical variability in their input parameters. In this approach, the reduced order system matrices are represented in terms of their orthogonal polynomial chaos expansions on the probability space defined by the input random variables. The coefficients of these polynomials, which are matrices, are obtained through the repeated, deterministic model order reduction of finite element models generated for specific values of the input random parameters. These values are chosen efficiently in a multi-dimensional grid using a Smolyak algorithm. The generated stochastic reduced order model is represented in the form of an augmented system that lends itself to the direct generation of the desired statistics of the device response. The accuracy and efficiency of the proposed method is demonstrated through its application to the reduced-order finite element modeling of a terminated coaxial cable and a circular wire loop antenna.

Improved gearbox simulations for diagnostic and prognostics purposes using finite element model reduction techniques

Lumped parameter models (LPMs) are widely utilised to predict the dynamic behaviour of mechanical systems such as gearboxes. LPM gives reasonable representation of the dynamics of the system if masses can be lumped at certain locations, such as gears, shafts, bearings, etc. LPM have the advantage of simulating the structure using a limited number of degrees-of-freedom (DOF), which facilitates studying the behaviour of gears and bearings in the presence of nonlinearities and geometrical faults. However, it is difficult to account for the casing flexibility in the LPM models which is an important consideration in the lightweight structures such as in aircraft applications and this results in poor spectral matching over a wide frequency range. In the case of continuous systems, where masses are distributed equally over the structure (gearbox casing), other methods, such as finite element analysis, are often used to study the behaviour of the structure. The use of finite element model (FEM) results in a large number of DOF, which in turn complicates simulating the whole system’s response to the presence of nonlinearities and to gears and bearing faults. This in turn limits the validity of the simulated results and restricts their later usage in the diagnostics and prognostics of the gears and bearings. This paper describes the use of dynamic reduction technique to reduce the FEM of a gearbox casing into manageable and well representative DOF of the casing. The reduced model of the casing is embedded with the LPM of the internals (shafts, gears and bearings), which was previously developed with the aid of Simulink and has the capability of capturing stiffness nonlinearities arising from gears and bearings and has also the capability of simulating geometrical faults (spalls) for both gears and bearings. This paper mainly deals with the simulation of localised and extended inner race bearing faults. The results show the improvements obtained through combining the reduced model of the casing with the LPM model and a much better correspondence with measured signals.

Model order reduction of finite element models: improved component mode synthesis

The finite element (FE) approach constitutes an essential methodology when modelling the elastic properties of structures in various research disciplines such as structural mechanics, engine dynamics and so on. Because of increased accuracy requirements, the FE method results in discretized models, which are described by higher order ordinary differential equations, or, in FE terms, by a large number of degrees of freedom (DoF). In this regard, the application of an additional methodology, referred to as the model order reduction (MOR) or DoF condensation, is rather compulsory. Herein, a reduced dimension set of ordinary differential equations is generated, i.e. the initially large number of DoF is condensed, while aiming to keep the dynamics of the original model as intact as possible. In the commercially available FE software tools, the static and the component mode syntheses (CMS) are the only available integrated condensation methods. The latter represents the state of the art generating well-correlated reduced order models (ROMs), which can be further utilized for FE or multi-body systems simulations. Taking into consideration the information loss of the CMS, which is introduced by its part-static nature, the improved CMS (ICMS) method is proposed. Here the algorithmic scheme of the standard CMS is adopted, which is qualitatively improved by adequately considering the advantageous characteristics of another MOR approach, the so-called improved reduction system method. The ICMS results in better correlated reduced order models in comparison to all the aforementioned methods, while preserving the required structural properties of the original FE model.

Krylov model order reduction of finite element approximations of electromagnetic devices with frequency‐dependent material properties

AbstractA methodology is presented for the Krylov subspace‐based model order reduction of finite element models of electromagnetic structures with material properties and impedance boundary conditions exhibiting arbitrary frequency dependence. The proposed methodology is a generalization of an equation‐preserving Krylov model order reduction scheme for methodology for second‐order, linear dynamical systems. The emphasis of this paper is on the application of this method to the broadband model order reduction of planar circuits including lossy strips of arbitrary thickness and lossy reference planes. In particular, it is shown that the proposed model order reduction methodology provides for the accurately modelling of the impact of the frequency dependence of the internal impedance per unit length of the thick lossy strip on the electromagnetic response of the stripline structure over a very broad, multi‐GHz frequency band, extending all the way down to frequencies in the DC neighbourhood. In addition, the application of the proposed methodology to the broadband modelling of electromagnetic coupling between strips on either side of a lossy ground plane is demonstrated. Copyright © 2007 John Wiley & Sons, Ltd.

New Structural Dynamic Condensation Method for Finite Element Models

A new iterative method is proposed for the reduction of finite element models arising in test-analysis correlation, model updating, finite element modeling, vibration control, structural dynamic optimization, and so on. Based on the modified eigenvalue equation and the eigenvalue shifting technique, two constraint equations for the dynamic condensation matrix, which relates the deformations associated with the master and slave degrees of freedom, are derived. Two iterative schemes are presented for solving the constraint equations. In the second constraint equation, because the dynamic condensation matrix has nothing to do with the eigenpairs of the reduced model, it is unnecessary to calculate them in every iteration. This makes the iterative scheme more computationally efficient than the usual scheme, especially when the number of the master degrees of freedom is large. The accuracy of eigenvalues and eigenvectors of the reduced model is examined in every iterative step. The comparison of the present method with some typical iterative schemes proposed in the past shows that the new one has the highest accuracy. Numerical examples are also presented to show the efficiency of the proposed method.

(M, N)-approximation: a system simplification method

A method is presented for the simplification of systems made up of N interacting subsystems, by approximating only M of the N subsystems. It is proposed to reduce each of the M subsystems while all interactions are active, and to preserve the identities of all subsystems in the overall approximation, as demanded in many situations. Problems of conventional model reduction, controller or compensator reduction, reduction of finite element models, reduction of decentralized controllers, etc., can be treated as specific applications of this method. As an application to controller reduction, a new version of balanced controllers is shown to be generated by this method. A numerical example is included that compares the new method with other existing balancing methods, and the results are shown to be in favour of the new method.

Simulation of Dynamics of Robots with Flexible Actuating Mechanisms

The problem of formalization of simulation of industrial robots dynamics is discussed with special attention devoted to taking into account of flexibility of links, Joints and transmission mechanisms. The model of actuating mechanism (manipulator) is constructed by ‘connection’ of models of separate flexible links, the movements of which are considered as deflections from nominal movement of a rigid mechanism. Special method is proposed for order reduction of finite element models of links by separating the appropriate systems of linear equations in ‘slow’ and ‘rapid’ parts. The transformation matrixes are found by iteration solving of corresponding Riccati equations. A short description of simulation algorithms and programs is given.