Finite Element Degrees Of Freedom Research Articles

FINITE ELEMENT METHOD FOR NONLINEAR EDDY CURRENT PROBLEMS IN POWER TRANSFORMERS

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady state periodic solution is of interest only. This is represented in the frequency domain as a Fourier series for each finite element degree of freedom and a finite number of harmonics is to be determined, i.e. a harmonic balance method is applied. Due to the nonlinearity, all harmonics are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a timeindependent permeability distribution, the so called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps resulting in two advantages. One is that each harmonic is obtained by solving a system of algebraic equations with only as many unknowns as there are finite element degrees of freedom. A second benefit is that these systems are independent of each other and can be solved in parallel. The appropriate selection of the fixed point permeability accelerates the convergence of the nonlinear iteration. The method is applied to the analysis of a large power transformer. The solution of the electromagnetic field allows the computation of various losses like eddy current losses in the massive conducting parts (tank, clamping plates, tie bars, etc.) as well as the specific losses in the laminated parts (core, tank shielding, etc.). The effect of the presence of higher harmonics on these losses is investigated.

On the High-Resolution Discretization of the Maxwell Equations in a Composite Tape and the Heating Effects Induced by the Dielectric Losses

Electromagnetic field propagation inside composite materials represents a challenge where fiber-scale simulation remains intractable using classical simulation methods. The present work proposes an original 3D simulation with a mesh resolution fine enough to resolve the fiber scale, thanks to the use of Proper Generalized Decomposition (PGD)-based space decomposition, which avoids the necessity of considering homogenized properties and considers the richest description of the involved physics from the solution of the Maxwell equations. This high-resolution simulation enables comparing the electromagnetic field propagation in a composite part, depending on the considered frequency and the fiber’s/wave polarization’s relative orientation. The electromagnetic fields are then post-processed to identify the heat generation terms and- the resulting induced thermal field. The results prove the ability of the PGD-based discretization to attain extremely high levels of resolution, the equivalent of 1010 finite-element degrees of freedom. The obtained results show an enhanced wave penetration when the electric field polarization coincides with the fiber orientation. On the contrary, when the electric field is polarized along the normal to the fiber orientation, both the penetration and the associated heating reduce significantly, compromising the use of homogenized models, rendering them unable to reproduce the observed behaviors.

Semihyper-Reduction for Finite Element Structures With Nonlinear Surface Loads on the Basis of Stress Modes

Abstract Model reduction via projection is a common method to accelerate time integration of finite element (FE) structures by reducing the number of degrees-of-freedom (DOFs). However, nonlinear state-dependent surface loads are usually computed based on the nonreduced DOFs of the FE model. When a considerably high number of DOFs are involved in the nonlinear surface loads, their computation becomes a bottleneck. This paper presents a general approach for reduced time integration and reduced force computation for FE models. The required force trial vectors can be computed easily and systematically out of deformation trial vectors, commonly called “modes.” Those force trial vectors, which we call “stress modes,” can be determined a priori so that a nonlinear computation of the full system is not necessary. The new idea in this contribution is that stress recovery is used to decrease the number of equations for the force computation. A general framework for semihyper-reduction (SHR) is developed and its practical implementation is discussed. The term SHR is introduced because it is an intermediate approach between the straight-forward method of using the FE DOFs and pure hyper-reduction (HR) where the FE DOFs are omitted for computing state-depended surface loads. In order to demonstrate the proposed SHR approach practically, a numerical example of a planar crank drive is given, where a hydrodynamic lubrication film separates piston and cylinder. Thereby, very good result quality has been observed in comparison to a finite difference reference solution.

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.

Advantages of the mixed format in geometrically nonlinear analysis of beams and shells using solid finite elements

SummaryThe paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path‐following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.

Exact microstructure-dependent Timoshenko beam element

In this study, we develop an exact microstructure-dependent Timoshenko beam finite element. First, a Timoshenko beam model based on the modified couple-stress theory is reviewed briefly. The general closed-form solution to the equilibrium equations of the beam is derived. The two-dimensional in-plane stress distributions along the beam in terms of load resultants are obtained. In addition, the general solution can be used to model any distributed load which can be expressed as a Maclaurin series. Then, the solution is presented in terms of discrete finite element (FE) degrees of freedom. This representation yields the exact shape functions for the beam element. Finally, the exact beam finite element equations are obtained by writing the generalized forces at the beam ends using the FE degrees of freedom. Three calculation examples which have applications in micron systems and sandwich structures are presented.

Full-field dynamic strain prediction on a wind turbine using displacements of optical targets measured by stereophotogrammetry

Health monitoring of rotating structures (e.g. wind turbines and helicopter blades) has historically been a challenge due to sensing and data transmission problems. Unfortunately mechanical failure in many structures initiates at components on or inside the structure where there is no sensor located to predict the failure. In this paper, a wind turbine was mounted with a semi-built-in configuration and was excited using a mechanical shaker. A series of optical targets was distributed along the blades and the fixture and the displacement of those targets during excitation was measured using a pair of high speed cameras. Measured displacements with three dimensional point tracking were transformed to all finite element degrees of freedom using a modal expansion algorithm. The expanded displacements were applied to the finite element model to predict the full-field dynamic strain on the surface of the structure as well as within the interior points. To validate the methodology of dynamic strain prediction, the predicted strain was compared to measured strain by using six mounted strain-gages. To verify if a simpler model of the turbine can be used for the expansion, the expansion process was performed both by using the modes of the entire turbine and modes of a single cantilever blade. The results indicate that the expansion approach can accurately predict the strain throughout the turbine blades from displacements measured by using stereophotogrammetry.

The modal reduction method for multi‐body dynamics with non‐smooth contact

SUMMARYThe elastic multi‐body system dynamics is studied from the prospect of modal reduction. The bodies are seen as stiff, close to rigid, subjected to non‐smooth contact interactions, such as collisions and dry friction. The finite element degrees of freedom of the bodies are represented either through the method of a floating frame of reference or the classical small deformation method in the global frame. The numerical modal reduction strategy based on Craig–Bampton representation of the current degrees of freedom is suggested. The method is general and may be applied to both quasi‐static and dynamic simulations. The key modeling assumption for the method is that the neglected modes have constant response in time. Numerical experiments, where the method is examined and compared with modally full floating frame approach, show satisfying accuracy while achieving significant computational savings. Copyright © 2014 John Wiley & Sons, Ltd.

A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

We consider a static condensation reduced basis element framework for efficient approximation of parameter-dependent linear elliptic partial differential equations in large three-dimensional component-based domains. The approach features an offline computational stage in which a library of interoperable parametrized components is prepared; and an online computational stage in which these component archetypes may be instantiated and connected through predefined ports to form a global synthesized system. Thanks to the component-interior reduced basis approximations, the online computation time is often relatively small compared to a classical finite element calculation. In addition to reduced basis approximation in the component interiors, we employ in this paper port reduction with empirical port modes to reduce the number of degrees of freedom on the ports and thus the size of the Schur complement system. The framework is equipped with efficiently computable a posteriori error estimators that provide asymptotically rigorous bounds on the error in the approximation with respect to the underlying finite element discretization. We extend our earlier approach for two-dimensional scalar problems to the more demanding three-dimensional vector-field case. This paper focuses on linear elasticity analysis for large structures with tens of millions of finite element degrees of freedom. Through our procedure we effectively reduce the number of degrees of freedom to a few thousand, and we demonstrate through extensive numerical results for a microtruss structure that our approach provides an accurate, rapid, and a posteriori verifiable approximation for relevant large-scale engineering problems.

Prediction of Full Field Dynamic Stress/Strain from Limited Sets of Measured Data

Transient events may cause degradation of a structure’s useful life. Often these events have an unknown effect on the structure. Some examples are helicopter hard landing, strong wind gusts on large turbine blades, and other similar occasional events. In many cases, only a handful of sensors are available for the evaluation of these rare, short-term events. This limited data does not always provide the detail necessary in order to determine the effects of the events that occur; this is very important especially in terms of the overall fatigue usage for the system. While a finite element model may be available, use of this limited data does not lend itself to easily integrate into the identification of the dynamic stress and dynamic strain. Often times, estimates of the forcing function can be approximated and used to identify the response. But these techniques are very approximate. A newly developed real time operating data expansion technique has been successfully used for the identification of more detailed information from limited sets of data. This expansion has mainly been used to augment the limited data acquired and allows for better overall visualization of these transient types of events. This technique has mainly been used for general response characteristics for structures. In this work, expansion techniques are developed in conjunction with the finite element mass and stiffness matrices to provide information at all of the finite element degrees of freedom from these limited sets of measured data points. With this approach, dynamic stress and dynamic strain can be identified from measured transient events thereby enabling estimation of fatigue accumulation or usage. Several cases are described to illustrate the overall process and results that are obtained from the expansion technique; these are compared to full reference solution results to verify the accuracy of the technique.

Sparse finite element methods for operator equations with stochastic data

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments $$\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,...,\mathcal{M}^k u \in V \otimes ... \otimes V$$ of the solution. We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment $$\mathcal{M}^k u$$ for k⩾1. The key tool in both algorithms is a “sparse tensor product” space for the approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom, instead of N k degrees of freedom for the full tensor product FEM space. A sparse Monte-Carlo FEM with M samples (i.e., deterministic solver) is proved to yield approximations to $$\mathcal{M}^k u$$ with a work of O(M N(log N) k−1) operations. The solutions are shown to converge with the optimal rates with respect to the Finite Element degrees of freedom N and the number M of samples. The deterministic FEM is based on deterministic equations for $$\mathcal{M}^k u$$ in D k ⊂ ℝkd. Their Galerkin approximation using sparse tensor products of the FE spaces in D allows approximation of $$\mathcal{M}^k u$$ with O(N(log N) k−1) degrees of freedom converging at an optimal rate (up to logs). For nonlocal operators wavelet compression of the operators is used. The linear systems are solved iteratively with multilevel preconditioning. This yields an approximation for $$\mathcal{M}^k u$$ with at most O(N (log N) k+1) operations.

Dynamic analysis of partially prismatic thin-walled structures

A mixed-mode method is investigated for the dynamic analysis of partially prismatic structures, such as box girders with intermediate diaphragms and supports, or with variable-depth webs. Thus the prismatic ‘main structure’ is discretised by finite strips, and the non-prismatic ‘sub-structure’ by finite elements. After dynamic reduction of the finite element degrees of freedom, advantage may be taken of the finite strip harmonic expansions in determining mode shapes. The method is tested by a variety of examples. Thus for structures with transverse substructures (e.g. diaphragms), the method is shown to be successful, being up to an order of magnitude faster than finite elements. However, for structures with longitudinal substructures (e.g. variable-depth webs) the method is shown to be unreliable. This is because for such structures, approximations in the dynamic reduction procedure are found to be significant.