Full-order Finite Element Model Research Articles

Transfer-learning-based strategy for enhancing prediction accuracy and computational efficiency of nonlinear mechanical properties in composite materials

In modern composite design, superior macroscopic mechanical properties can be achieved by optimizing the microstructures of materials. However, the direct prediction of the microstructure–property relationship under nonlinear conditions through numerical methods can be inefficient or imprecise. In this study, a transfer learning strategy based on the reduced order model (ROM) was proposed, offering an enhanced and rapid prediction of the mechanical responses of composite materials under nonlinear conditions. Initially, an extensive training dataset was generated by applying the ROM, used to pre-train the neural network model. Following this, a few data calculated from full-order finite element models were leveraged to fine-tune the network parameters. This approach exploits the efficient data generation capability of the ROM, with its potential computational inaccuracies in nonlinear scenarios mitigated, leading to an improvement in the accuracy and efficiency of the surrogate model. Numerical examples of a nonlinear hyper-elastic material with inclusions were examined, revealing that the computational cost in the offline stage of the transfer learning method is only half that of traditional neural network models, and it enable near real-time predictions in the online stage. Notably, it was shown that the accuracy loss of the developed surrogate model in scenarios of strong non-linearity is significantly less than that of the ROM. This method presents an innovative pathway for the swift and accurate evaluation of the effective mechanical properties of composite structures, with the potential to offer valuable insights for related methodological research.

Reduced left ventricular dynamics modeling based on a cylindrical assumption.

Biomechanical modeling and simulation is expected to play a significant role in the development of the next generation tools in many fields of medicine. However, full-order finite element models of complex organs such as the heart can be computationally very expensive, thus limiting their practical usability. Therefore, reduced models are much valuable to be used, for example, for pre-calibration of full-order models, fast predictions, real-time applications, and so forth. In this work, focused on the left ventricle, we develop a reduced model by defining reduced geometry & kinematics while keeping general motion and behavior laws, allowing to derive a reduced model where all variables & parameters have a strong physical meaning. More specifically, we propose a reduced ventricular model based on cylindrical geometry & kinematics, which allows to describe the myofiber orientation through the ventricular wall and to represent contraction patterns such as ventricular twist, two important features of ventricular mechanics. Our model is based on the original cylindrical model of Guccione, McCulloch, & Waldman (1991); Guccione, Waldman, & McCulloch (1993), albeit with multiple differences: we propose a fully dynamical formulation, integrated into an open-loop lumped circulation model, and based on a material behavior that incorporates a fine description of contraction mechanisms; moreover, the issue of the cylinder closure has been completely reformulated; our numerical approach is novel aswell, with consistent spatial (finite element) and time discretizations. Finally, we analyze the sensitivity of the model response to various numerical and physical parameters, and study its physiological response.

Efficient Method for Derivatives of Nonlinear Stiffness Matrix

Structural design often includes geometrically nonlinear analysis to reduce structural weight and increase energy efficiency. The full-order finite element model can perform the geometrically nonlinear analysis, but its computational cost is expensive. Therefore, nonlinear reduced-order models (NLROMs) have been developed to reduce costs. The non-intrusive NLROM has a lower cost than the other due to the approximation of the nonlinear internal force by a polynomial of reduced coordinates based on the Taylor expansion. The constants in the polynomial, named reduced stiffnesses, are derived from the derivative of the structure’s tangential stiffness matrix with respect to the reduced coordinates. The precision of the derivative of the tangential stiffness affects the reduced stiffness, which in turn significantly influences the accuracy of the NLROM. Therefore, this study evaluates the accuracy of the derivative of the tangential stiffness calculated by the methods: finite difference, complex step, and hyper-dual step. Analytical derivatives of the nonlinear stiffness are developed to provide references for evaluating the accuracy of the numerical methods. We propose using the central difference method to calculate the stiffness coefficients of NLROM due to its advantages, such as accuracy, low computational cost, and compatibility with commercial finite element software.

Constructing simplified models for dynamic analysis of monopile-supported offshore wind turbines

Accurately modeling the complicated dynamic mechanisms of offshore wind turbines (OWTs) requires high numerical fidelity. However, since these models are computationally expensive for both forward and inverse problems, simplified OWT models are needed. The present study develops two approaches to constructing simplified numerical models based on full-order finite element models, which aim to accurately and efficiently simulate dynamic behaviours of monopile-supported OWTs. The first approach decomposes the structure into substructures and establishes a reduced-order model based on the similarity principle of modal parameters and time-domain responses, in accordance with full-order models. The second approach uses proper orthogonal decomposition and modal truncation techniques to construct a state-space model, which produces consistent time-domain responses with full-order models. For both approaches, model updating is used to improve the accuracy of the simplified models. The effectiveness of the approaches is demonstrated through the monopile-supported, 10 MW DTU reference wind turbine. The results show that the computational time required for both simplified models is less than 10% that of the full-order model, without the compromise of accuracy, i.e., less than 5% difference in modal frequency and an excellent agreement in time-domain responses. The proposed approach provides an efficient and accurate dynamic analysis solution for offshore structures.

Combining Reduced-Order Stick Model with Full-Order Finite Element Model for Efficient Analysis of Self-Elevating Units

Reduced-order stick models are frequently employed to obtain dynamic amplification factors of self-elevating units (SEU), while the full-order finite element (FE) models are used for quasi-static analyses. This paper develops an efficient framework to create structural digital twins for SEUs by combining both stick models and full-order FE models. A stick model and a detailed FE model of an in-house developed generic SEU are established, respectively, following the standard industry guideline. Dynamic analyses are performed for the stick model based on the modal superposition method. Assuming that the stick model contains the key dynamic characteristics of the full-order FE model, the modal participation factors are multiplied by the corresponding mode shapes of the full FE model to derive the global dynamic responses of the entire SEU. The derived nodal displacements are imposed on the full-order model to obtain the member stresses. The global responses and member stresses are benchmarked with the results from a direct full-order dynamic FE analysis for various environmental conditions. The presented framework is found to significantly increase the efficiency of the simulation while retaining a similar accuracy, and it forms a critical step for the ongoing development of digital twins of fixed offshore structures.

A Gaussian process regression reduced order model for geometrically nonlinear structures

Reduced order models, such as Hollkamp and Gordon’s Implicit Condensation and Expansion (ICE) model, are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. However, a reduced order model (ROM) is typically only valid for one FEM. It does not capture how each ROM coefficient changes due to variations in the FEM (e.g., design parameters or uncertainties), so if the FEM is updated then the ROM needs to be re-computed with a new set of static load–displacement data. This study presents a data-driven reduced order modeling approach that creates a single ROM that incorporates design variations in FEM. The proposed method applies Gaussian Process Regression (GPR) to the ICE approach, making each coefficient in an ICE ROM a regression model with respect to a collection of FEMs with varying material properties or geometric parameters. Once the GPR ROM has been identified, one can immediately produce an ICE ROM for a set of FEM parameters without a need to solve any static load–displacement cases on the full FEM. This dramatically enhances the computational efficiency and could be helpful when model uncertainty needs to be considered or when seeking to update a model to correlate with measurements. Additionally, the coefficients of a ROM can often change considerably if the scale on the load–displacement data changes, so it can be difficult to know whether the scaling that was chosen has really identified an accurate ROM. The proposed GPR ROM estimates the mean ROM coefficients for a range of load scaling as well as the uncertainty on each ROM coefficient with respect to the load level. This can be used to gauge the success of the ROM identification and to eliminate ROM coefficients that are unimportant and hence highly variable. The proposed GPR ROM approach is evaluated by applying it to flat and curved beam structures, revealing that the advantages outlined above can be realized with a relatively modest increase in cost relative to a traditional ICE ROM.

Real-time prediction by data-driven models applied to induction heating process

Data-driven modeling approach constitutes an appealing alternative to the finite element method for optimizing complex multiphysics parametrized problems. In this context, this paper aims at proposing a parametric solution for the temperature-time evolution during the multiphysics induction heating process by using a data-driven non-intrusive modeling approach. To achieve this goal, firstly, a set of synthetic solutions was collected, at some sparse sensors in the space domain and for properly selected process parameters, by solving the full-order finite element models using FORGE® software. Then, the gappy proper orthogonal decomposition method was used to complete the missing data. Next, the proper orthogonal decomposition method coupled with the nonlinear sparse proper generalized decomposition regression method was applied to find a low-dimensional space onto which the original solutions were projected and a model for the low-dimensional representations was, therefore, created. Hence, a real-time prediction of the temperature-time evolution and for any new process parameters could be efficiently computed at the predefined positions (sensors) in the space domain. Finally, spatial interpolation was carried out to extend the solutions everywhere in the spatial domain by applying a strategy based on the nonlinear dimensionality reduction by locally linear embedding method and the proper orthogonal decomposition method with radial basis functions interpolation. It was shown that the results are promising and the applied approaches provide good approximations in the low-data limit case.

Component-wise reduced order model lattice-type structure design

Lattice-type structures can provide a combination of stiffness with light weight that is desirable in a variety of applications. Design optimization of these structures must rely on approximations of the governing physics to render solution of a mathematical model feasible. In this paper, we propose a topology optimization (TO) formulation that approximates the governing physics using component-wise reduced order modeling as introduced in Huynh et al. (2013); Eftang and Patera (2013), which can reduce solution time by multiple orders of magnitude over a full-order finite element model while providing a relative error in the solution of <1%. In addition, the offline training data set from such component-wise models is reusable, allowing its application to many design problems for only the cost of a single offline training phase, and the component-wise method is nearly embarrassingly parallel. We also show how the parameterization chosen in our optimization allows a simplification of the component-wise reduced order model (CWROM) not noted in previous literature, for further speedup of the optimization process. The sensitivity of the compliance with respect to the particular parameterization is derived solely at the component level. In numerical examples, we demonstrate a 1000x speedup over a full-order FEM model with relative error of <1% and show minimum compliance designs for two different cantilever beam examples, one smaller and one larger. Finally, error bounds for the displacement field, compliance, and compliance sensitivity of the CWROM are derived.

A Study of Whole Joint Model Calibration Using Quasi-Static Modal Analysis

Abstract Small length and time scales resulting from high-fidelity frictional contact elements make long duration, low frequency simulations intractable. Alternative reduced order modeling approaches for structural dynamics models have been developed over the last several decades to approximate joint physics based on empirical or mathematical models within a whole joint model representation. The challenge with nonlinear constitutive elements based on empirical models is that the parameters must be calibrated to either experimental or simulation data. This research proposes a model calibration technique that identifies the joint parameters of a four-parameter Iwan element based on the nonlinear natural frequencies and damping ratios computed with quasi-static modal analysis (QSMA). The QSMA algorithm is applied to the full-order finite element model (FEM) to obtain reference data, and a genetic algorithm optimizes the joint parameters within a reduced order model (ROM) by minimizing the difference between the nonlinear modal characteristics for the modes of interest. The calibration method is demonstrated on a C-Beam bolted assembly and the resulting reduced order model is validated by comparing simulations of broadband, forced transient response. The resulting calibrated model captures the nonlinear, multimodal response at a significantly reduced computational cost and can be utilized for producing efficient models that do not have supporting experimental data for calibration.

An extended subset of nominal modes method for coated blisks with thickness mistuning of coatings and blisks

The thickness mistuning of coatings and blisks cannot be predicted before manufacturing process, and the mistuning patterns normally tend to be random, which has not been investigated in opened studies. A novel finite element model (FEM) is developed to incorporate variable thicknesses of the coatings and the blisks. The FEM requires only the sector-level node coordinates of the uncoated blisks and is validated by the ANSYS software. Nevertheless, the classical subset of nominal modes (SNM) cannot capture the accuracy of coated blisks with thickness mistuning and to solve the eigenproblem of the full-order FEM will require expensive computational cost. As a consequence, on the basis of a subspace iteration method, an extended SNM is developed to improve the computational efficiency, where the initial mode matrix of the iteration is obtained by using the Craig–Bampton method and the cyclic symmetric boundary conditions. This paper presents a numerical comparison of the classical SNM, the extended SNM and the full-order FEM. The results suggest that the developed FEM and the extended SNM are effective for the analysis with thickness mistuning of the coatings and the substrate. Compared with the solution time of the eigenproblem for the full-order FEM, it is reduced by roughly 89.5% for the extended SNM.

Reduced Order Models for Nonlinear Dynamic Analysis With Application to a Fan Blade

Abstract In this work, reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical nonlinearities are developed and applied to the dynamic study of a fan. The structure is considered to present nonlinear vibrations around the prestressed equilibrium induced by rotation enhancing the classical linearized approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the stiffness evaluation procedure (STEP) and then corrected by means of a proper orthogonal decomposition (POD) that filters the full order nonlinear forces (StepC ROM). The linear normal modes (LNM) and Craig-Bampton (C-B) type reduced basis are considered here. The latter are parameterized with respect to the rotating velocity. The periodic solutions obtained with the StepC ROM are in good agreement with the solutions of the FOM and are more accurate than the linearized ROM solutions and the STEP ROM. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM.

Two-tier model reduction of viscoelastically damped finite element models

Transient simulations of linear viscoelastically damped structures require excessive computational resources to directly integrate the full-order finite element model with time-stepping algorithms. Traditional modal reduction techniques are not directly applicable to these systems since viscoelastic materials depend on time and frequency. A more appropriate reduction basis is obtained from the nonlinear, complex eigenvalue problem, whose eigenvectors capture the appropriate kinematics and enable frequency-based mode selection; unfortunately, the computational cost is prohibitive for computing these modes from large-scale engineering models. To address this shortcoming, this work proposes a novel two-tier reduction procedure to reduce the upfront cost of solving the complex, nonlinear eigenvalue problem. The first reduction step reduces the full-order model with real mode shapes linearized about various centering frequencies to capture the kinematics over a full range of viscoelastic material behavior (glassy, rubbery, and glass-transition zones). This tier-one reduction preserves time-temperature superposition and allows the equations to depend parametrically on operating temperature. The second-level reduction then solves the complex, nonlinear eigenmode solutions in the tier-one reduced space about a fixed temperature to further reduce the equations-of-motion. The method is demonstrated on a cantilevered sandwich plate to showcase its accuracy and efficiency in comparison to full-order model predictions.

Orthogonal Interpolation Method for Order Reduction of a Synchronous Machine Model

This paper introduces an interpolation method based on snapshot approach to reduce the order of a nonlinear model for magnetostatic problems and facilitate the evaluation of the corresponding system of equations. The solution of the reduced model for a set of given inputs is built by interpolating the right singular vectors, obtained from singular value decomposition of snapshots. To validate the efficiency of this method, this reduced model is compared with full-order finite-element model. Furthermore, the proposed method is analyzed with respect to a more conventional model order reduction that is combined with the discrete empirical interpolation method. The orthogonal interpolation method is the most effective method to reduce model of an interior permanent magnet synchronous machine especially in term of computational time.

Model refinement for offshore platforms: Experimental study

Offshore jacket platforms are widely used in offshore oil and gas exploitation. Finite element models of such structures need to have many degrees of freedom (DOFs) to represent the geometrical detail of complex structures, thereby leading to incompatibility in the number of DOFs of experimental models. To bring them both to the same order while ensuring that the essential eigen- properties of the refined model match those of experimental models, an extended model refinement procedure is presented in this paper. Vibration testing of an offshore jacket platform model is performed to validate the applicability of the proposed approach. A full-order finite element model of the platform is established and then tuned to meet the measured modal properties identified from the acceleration signals. Both model reduction and modal expansion methods are investigated, as well as various scenarios of sensor arrangements. Upon completion of the refinement, the updated jacket platform model matches the natural frequencies of the measured model well.

Dynamic Response Analysis of Elastic-plastic Structures with Component Mode Synthesis

This paper evaluates the validity of component mode synthesis (CMS) technique for dynamic response of elastic-plastic structures. A method incorporating elastic-plastic material behaviour and the fixed interface mode synthesis technology is presented. To do this, the incremental governing equations of elastic-plastic transient problem derived from nonlinear finite element theory are linearized in each time step, the tangent stiffness matrix is used to generate the basis vector of transformation matrix. Using the proposed method, the response of a simply supported beam loaded impulsively is examined. The Numerical results indicate that the proposed method is accurate, convergent and more efficient compared with the full order finite element model.

Finite element model calibration of a nonlinear perforated plate

This paper presents a case study in which the finite element model for a curved circular plate is calibrated to reproduce both the linear and nonlinear dynamic response measured from two nominally identical samples. The linear dynamic response is described with the linear natural frequencies and mode shapes identified with a roving hammer test. Due to the uncertainty in the stiffness characteristics from the manufactured perforations, the linear natural frequencies are used to update the effective modulus of elasticity of the full order finite element model (FEM). The nonlinear dynamic response is described with nonlinear normal modes (NNMs) measured using force appropriation and high speed 3D digital image correlation (3D-DIC). The measured NNMs are used to update the boundary conditions of the full order FEM through comparison with NNMs calculated from a nonlinear reduced order model (NLROM). This comparison revealed that the nonlinear behavior could not be captured without accounting for the small curvature of the plate from manufacturing as confirmed in literature. So, 3D-DIC was also used to identify the initial static curvature of each plate and the resulting curvature was included in the full order FEM. The updated models are then used to understand how the stress distribution changes at large response amplitudes providing a possible explanation of failures observed during testing.

A Generalized Iterative Approach to Improve Reduced-Order Model Accuracy for Inverse Problem Applications

AbstractA generally applicable algorithm for the iterative generation of data ensembles to efficiently create accurate computational mechanics reduced-order models (ROMs) for use in computational approaches to approximate inverse problem solutions is presented and numerically evaluated. The ROM approach considered is based on identifying the optimal low-dimensional basis to be used within a Galerkin weak-form finite-element (FE) method to provide substantially reduced computational cost while maintaining accuracy relative to that of a (traditional) full-order FE model. Furthermore, proper orthogonal decomposition is used to derive the ROM basis from a set of response fields (i.e., snapshots) generated a priori with full-order FE analyses. Therefore, the set of full-order FE analyses used to create the ROM directly affects the accuracy/generalization of the ROM. The core hypothesis of the algorithm presented is that maximizing the diversity, as defined in a measurable sense, of the full-order models (FOM) ...

A frequency-based substructuring approach to efficiently model position-dependent dynamics in machine tools

Structural deformations of machine tool components that translate and/or rotate relative to each other to realize tool motion results in tool point dynamics varying along the tool path. These changing dynamics interact with the cutting process and the control loop of the drives to limit machine performance, making it necessary to virtually characterize these interactions such as to guide design decisions. To facilitate rapid evaluation of these varying dynamics, this paper describes a generalized frequency-based substructuring approach that combines the position-invariant component level receptances at the contacting interfaces between substructures to obtain the position-dependent tool point response. Receptances at the contacting interfaces are approximated by projecting them to a point to facilitate a multiple point receptance coupling formulation. Complete machine behavior is represented by just a few sets of receptances, making the model computationally more efficient than full-order finite element models and other dynamic substructuring methods. Position-dependent dynamic behavior for a representative three axis milling machine is simulated and numerically verified. Rapid investigations of the varying dynamics assist in virtually characterizing machine performance before eventual prototyping.

Modeling vibratory damage with reduced-order models and the generalized finite element method

This paper investigates a coupled computational analysis framework that uses reduced-order models and the generalized finite element method to model vibratory induced stress near local defects. The application area of interest is the life prediction of thin gauge structural components exhibiting nonlinear, path-dependent dynamic response. Full-order finite element models of these structural components can require prohibitively large amounts of processor time. Recent developments in nonlinear reduced-order models have demonstrated efficient computation of the dynamic response. These models are relatively insensitive to small imperfections. Conversely, the generalized finite element method provides the ability to model local defects without geometric dependency on the mesh. A more robust version of the method, with numerically built enrichment functions, provides a multiple-scale modeling capability through direct coupling of global and local finite element models. Replacing the component finite element model with a reduced-order model allows for efficient computation of dynamic response while providing the necessary information to drive local, solid analyses which can zoom in on regions containing stress risers or cracks. This paper describes the coupling of these approaches to enable fatigue and crack propagation predictions. Numerical/experimental examples are provided.

Parametric reduced-order models of battery pack vibration including structural variation and prestress effects

The goal of this work is to develop a numerical model for the vibration of hybrid electric vehicle (HEV) battery packs to enable probabilistic forced response simulations for the effects of variations. There are two important types of variations that affect their structural response significantly: the prestress that is applied when joining the cells within a pack; and the small, random structural property discrepancies among the cells of a battery pack. The main contributions of this work are summarized as follows. In order to account for these two important variations, a new parametric reduced order model (PROM) formulation is derived by employing three key observations: (1) the stiffness matrix can be parameterized for different levels of prestress, (2) the mode shapes of a battery pack with cell-to-cell variation can be represented as a linear combination of the mode shapes of the nominal system, and (3) the frame holding each cell has vibratory motion. A numerical example of an academic battery pack with pouch cells is presented to demonstrate that the PROM captures the effects of both prestress and structural variation on battery packs. The PROM is validated numerically by comparing full-order finite element models (FEMs) of the same systems.