Finite Element Matrix Research Articles

Further research of multi-scale assemble and matrix reassemble method for efficient analysis of railway track–substructure system subjected to a moving train

Abstract For a large-scale dynamic system, the efficiency of computation becomes a vital work sometimes in engineering practices. As a layered structural system, ballastless track and substructure occupy most part of the degrees of freedom of the whole system. It is, therefore, rather important to optimize the structural models in dynamic equation formulations. In this work, a three-dimensional and coupled model for multi-rigid-body of train and finite elements of track and substructures is presented by multi-scale assemble and matrix reassemble method. The matrix reassembling tactic is based on the multi-scale assemble method, through which the finite element matrix bandwidth is greatly narrowed, and the Cholesky factorization, iterative and multi-time-step solution have been introduced to efficiently obtain the train, track and substructure responses. The subgrade and its subsoil works as a typical substructural system, and comparisons with the previous model without matrix reassembling, SIMPACK and ABAQUS have been conducted to fully validate the efficiency and accuracy of this train–track–subgrade dynamic interaction model.

The H1-error analysis of the reduced-order extrapolated method for parabolic equation

Abstract In this study, we focus on the proper orthogonal decomposition reduced-order extrapolated technique in the backward Euler finite element (BEFE) method applied to parabolic equation to improve the solving efficiency. We begin by presenting the BEFE method for our given problem, including the existential stability and the H 1-error estimates about the BEFE numerical approximation. Next, we construct the reduced-order extrapolated finite element (ROEFE) matrix model based on the proper orthogonal decomposition reduced-order extrapolated technique. We then prove the existential stability and H 1-norm error estimates for the ROEFE model solutions using norm theory. Our results demonstrate that the precision of the ROEFE model matches that of the BEFE method in the same order, while significantly reducing CPU time under the same time step. Lastly, the numerical case verifies the feasibility and superiority of the ROEFE matrix model over traditional methods.

Analytical, computational, and experimental investigations on the impact of side outlet on the simple expansion chamber with axial outlet

In this study, an investigation on the acoustic performance of a simple expansion chamber muffler with side outlet has been presented. The investigation for the muffler is done in the 10–2400 Hz frequency range. The muffler is analyzed analytically, computationally, and experimentally. The computational and analytical modellings of the muffler have been done by using the finite element and transfer matrix methods, respectively for the determination of transmission loss. As the transfer matrix method is unable to capture the higher order modes above the cut-off frequency of the muffler, the finite element method has been adopted as the computational method. Additionally, the accuracy in the analytical and computational modelling of the muffler is checked with the experimental investigation. The two-load method is used in this study for muffler to determine the transmission loss and band power. The side outlet of the muffler introduces an extra impedance due to the shunt element and hence the effectiveness of the muffler can be improved. This muffler shows an overall improvement of 24.73 dB in the transmission loss when compared with the transmission loss of the simple expansion chamber muffler with an axial outlet in 10–2400 Hz frequency range. Specifically, the simple expansion chamber muffler with side outlet is more effective than the simple expansion chamber muffler with an axial outlet in 350–680 Hz, 1040–1350 Hz, 1570–1590 Hz, and 1850–2400 Hz frequency ranges. The comparison among the different techniques shows a fair agreement in the results. The transmission loss, variation of the band power in the 1/3rd octave band, and sound pressure level contours for the muffler at the selected frequencies have been presented. This study offers an advantage in the design of the automotive mufflers.

Blade reduced model considering local contact and analyzing blade vibration characteristics

A dynamic model of a tenoned rotating blade considering the pressure distribution characteristics of the contact interface is established and the degree of freedom of the model is reduced by the double coordination free-interface modal synthesis method in this paper. Firstly, the finite element model of the blade is established by ANSYS and the finite element matrix is extracted. Subsequently, multiple contact pairs were established on the contact surface of the tenon to capture the micro-slip characteristics of the contact interface. The normal pressure of the contact pair is obtained by the ANSYS blade-disk contact prediction and load equivalence to take the pressure distribution characteristics into account in the blade dynamics analysis. Then, non-uniform aerodynamic excitation is applied to the blade to simulate the load on the blade under real working conditions. Finally, the double coordination free-interface modal synthesis method is used to reduce the degree of freedom. Due to the coordination condition of the force and displacement of the substructure connection interface, the degree of freedom of the substructure interface is directly reduced. The final reduced degree of freedom is only the sum of the retained modes of the substructure, which greatly improves the computational efficiency. The dynamic equation is solved by the time integration method, and the effects of the tenon angle, rounded corner, and friction coefficient are discussed. The results show that the double coordination free-interface modal synthesis method can greatly reduce the degree of freedom of the tenon-connected blade model under the premise of ensuring accuracy while retaining the local contact characteristics of the contact interface. The structural parameters of the dovetail structure have a great influence on the vibration reduction characteristics of the blade. When designing the tenon structure, it is necessary to pay attention to these parameters.

Finite-Element A-V Formulation for EMQS Problems via Two-Domain Continuity Gauging

A finite-element A-V formulation is developed to solve EMQS (electromagnetic quasistatic), aka Darwin, problems in the time and frequency domains by making use of a two-domain continuity gauging strategy. The formulation utilizes the magnetic vector and the electric scalar potentials, and an additional scalar in the conducting regions to make the discretized Galerkin system symmetric. The resulting FE (finite element) matrix is LF (low-frequency) stable in the frequency domain. As a result, the time-step size is not limited by the characteristics of the system matrix in the backward Euler time-stepping scheme. The system matrixes are also regular, which allows choosing a direct solver to speed up the temporal solution significantly. The formulation is verified in both the frequency and the time domains by solving benchmark problems, and the results are compared to full-wave and eddy-current solutions.

Stiffness Modeling and Deformation Analysis of Parallel Manipulators Based on the Principal Axes Decomposition of Compliance Matrices

Abstract This paper presents a general equivalent approach to solve the stiffness modeling or load-deformation problem of non-redundant parallel mechanisms. Based on the principal axes decomposition of structure compliance matrices, an equivalent six-degrees-of-freedom (6-DOFs) serial mechanism is established to approximate the load-deformation behavior of each flexible link in the mechanism. Hence, each limb of the parallel mechanism can be equivalent to a serial redundant rigid body mechanism with passive elastic joints, and the load-deformation problem can be transformed to the equilibrium configuration calculation of the equivalent mechanism. The main advantage of the proposed method is that the robotic kinematics and statics, rather than the elastic mechanics, can be directly adopted to solve the equilibrium configuration of the parallel mechanism under external load. Besides, a closed form solution of the corresponding deformation can be obtained, which can be solved by the gradient-based searching algorithm. Therefore, the final deformation will no longer be linear to the external load, which makes this method more accurate and more suitable for the deformation prediction and compensation in real industrial working conditions. In order to verify the effectiveness and correctness of this method, a 3PRRU parallel manipulator will be introduced as an example, to compare the load-deformation results with the finite element analysis (FEA) simulation and matrix calculation methods, so the nonlinearity feature can be shown in an intuitive manner.

Aeroelastic stability analysis of rotor blade with complex three-dimensional design

A rotor dynamic analysis method for the complex three-dimensional design blade was developed and applied to analyzing its aeroelastic stability. Based on the medium-size deformation beam theory and Hamilton principle, the joint transfer matrix was used in the blade kinematic and the deformation compatibility principle was used in the assembled finite element matrix to develop the complex three-dimensional rotor structural dynamics model. The BO105 rotor was used to validate this method and the aeroelastic characteristic of the complex three-dimensional rotor was analyzed in detail. The results showed that the negative structural coupling of flap-torsion was appeared with tip sweep, and the 1/rev torsion modal frequency was decreased, and the 1/rev torsion modal damping ratio was maximally decreased about 90%. The positive structural coupling of lag-torsion was appeared with tip droop, the 2/rev lag modal frequency was decreased, the 1/rev torsion modal damping ratio was maximally decreased about 62%. The torsion stability was sharply decreased with tip sweep and droop design.

Accelerating high-order mesh optimization using finite element partial assembly on GPUs

In this paper we present a new GPU-oriented mesh optimization method based on high-order finite elements. Our approach relies on node movement with fixed topology, through the Target-Matrix Optimization Paradigm (TMOP) and uses a global nonlinear solve over the whole computational mesh, i.e., all mesh nodes are moved together. A key property of the method is that the mesh optimization process is recast in terms of finite element operations, which allows us to utilize recent advances in the field of GPU-accelerated high-order finite element algorithms. For example, we reduce data motion by using tensor factorization and matrix-free methods, which have superior performance characteristics compared to traditional full finite element matrix assembly and offer advantages for GPU-based HPC hardware. We describe the major mathematical components of the method along with their efficient GPU-oriented implementation. In addition, we propose an easily reproducible mesh optimization test that can serve as a performance benchmark for the mesh optimization community.

Theory and numerical analysis for two-dimensional recoil of deepwater drilling riser after emergency disconnection

Riser systems usually encounter large offset and huge environmental loads when emergency disconnection. However, the two-dimensional recoil theory and research still lack. This paper consequently establishes the two-dimensional riser recoil dynamics theory and two-dimensional mud discharge theory. The interaction mechanism between the riser and discharged mud is discussed, and the types of discharged load and the contribution of internal fluid (FI) to the riser finite element matrix are clarified. Combining the method of integration and coordinates reconstruction, the actual ordinate of the riser and elongation of the tensioner are retrieved. Numerical analysis results show the FI mass and gravity should not be included in the axial part of the riser element matrix during recoiling. The additional inertial load of FI due to the movement of the riser element influences the transverse displacement. The extremums of VM and bending moment appear at two ends of the riser system during early recoil stage. Transverse deformation of the riser system significantly affects the longitudinal displacement, which cannot be ignored. The dynamic responses of recoil in different two-dimensional scenarios are explored, including in different wave and current environments, at different vessel offset points, and with different vessel transverse velocities.

Vibration analysis of wedge beam with variable thickness based on energy finite element method

The variable thickness beam structure is widely used in various engineering structures due to its superior mechanical properties. However, the problem of high-frequency vibration is also quite obvious. Energy Finite Element Analysis (EFEA) is very effective for high frequency vibration response analysis. In order to apply the energy finite element analysis to accurately predict the high-frequency vibration response of a wedge beam structure with variable thickness, in this paper, the energy flow model of wedge beam with variable thickness is established. Firstly, the governing equation of energy density of wedge beam with variable thickness is derived by geometric acoustic approximation method, and the energy finite element matrix equation of the obtained governing equation of energy density is obtained by Galerkin variational method. The energy transfer characteristics of the structure coupled with the uniform beam at the coupling boundary are studied. Based on the energy transfer coefficient, the energy transfer relationship at the coupling boundary is established, and the overall finite element equation of the coupled beam structure is further obtained. The established analysis model can be used to analyze and predict the energy density of the wedge beam structure and the structure coupled with uniform beam.

Prediction and uncertainty quantification of the diffuse sound absorption of finite absorbers

This contribution presents a numerical approach to predict diffuse sound absorption. Conventionally, this is done by considering an infinite absorber coupled to an acoustic halfspace. However, the diffuse absorption coefficient for a finite absorber can be quite different from the corresponding infinite-size value due to what is referred to in literature as the size or edge effect. A finite size correction has been developed previously, but it is only applicable to homogeneous absorbers, neglects boundary conditions, and is based on a computationally costly numerical integration. This contribution presents an alternative approach which overcomes these drawbacks. In this approach, a deterministic model of the absorber, e.g., constructed with the finite element or modal transfer matrix method, is coupled to a diffuse model of the room through the diffuse field reciprocity relation. The theoretical uncertainty on this diffuse sound absorption that is inherent in the diffuse field assumption can also be quantified, or more precisely, the variance of the sound absorption that can be theoretically expected across a diverse ensemble of rooms of the same volume. It is demonstrated that the diffuse absorption coefficient corresponds to the average of the absorption coefficient across an ensemble of rooms, implying that the former is of practical use even at low frequencies. The methodology is experimentally validated for porous and membrane absorbers.

Optimal Design of Sliding Bearings Based on Artificial Intelligence Algorithm and CFD Simulation

Sliding bearings have a long history, simple manufacturing, and low cost. However, the function of sliding bearings is very large. As a key component of rotating machinery, thrust sliding bearing is directly related to the overall performance of the machine. The machines that can be seen in life, whether simple or complicated, have been used. However, with the development of science and technology and economy, all kinds of new things have been inoculated. One of the current issues is how sliding bearings can continue to keep machinery stable. CFD is the abbreviation of English computational fluid dynamics (computational fluid dynamics). It is developed with the development of computer technology and numerical computing technology. Simply put that CFD is equivalent to “virtually” doing experiments on the computer to simulate the actual fluid flow. The purpose of this article is to study the optimal design of sliding bearings by artificial intelligence algorithms and CFD simulation. This paper uses intelligent design methods to optimize the appearance, shape, and manufacturing process of sliding bearings and to carry the bearing capacity, friction coefficient and temperature rise of sliding bearings. Comprehensive research then establishes the objective function. The finite element algorithm, genetic algorithm, and matrix theory are used to calculate the stability of the sliding bearing under various conditions. The artificial intelligence algorithm is used to sort the calculated data. The CFD simulation is used to obtain the most reasonable result. Algorithm, as well as the appearance, shaft diameter, and material of the plain bearing, maintains stability under various conditions. The experimental data show that the classification using artificial intelligence algorithms and CFD simulation can significantly improve the performance of sliding bearings. It is found that the relationship between the sliding bearings is related to the speed, the diameter of the bearing, and other factors. Experimental data show that artificial intelligence algorithms and CFD simulation can provide reliable data references for the optimal design of sliding bearings, and the optimized sliding bearings can meet the stability under relevant conditions.

Investigation of electromagnetic wave propagation in the bicomplex 3D-FEM using a wavenumber Whitney Hodge operator

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

A Reduced-Order Extrapolation (ROE) Method for Solution Coefficient Vectors in the Mixed Finite Element (MFE) Method for the Two-Dimensional (2D) Fourth-Order Hyperbolic Equation

This study focuses on a reduced-order extrapolation method for the coefficient vectors of the mixed finite element solution for two-dimensional fourth-order hyperbolic equation. We first establish the mixed finite element scheme for the equation and give the matrix model of the mixed finite element scheme and the existence, stability and error estimates of its solutions. Then, we derive a reduced-order extrapolation mixed finite element matrix model with a small number of unknowns, where the proper orthogonal decomposition method is used to save central processing unit time, and prove the existence, stability and error estimates of the reduced-order extrapolation mixed finite element solutions with the help of matrix knowledge. More importantly, the reduced-order extrapolation mixed finite element matrix model have the same basis functions and error accuracy as the mixed finite element matrix model. Finally, some numerical experiments confirm the effectiveness of the reduced-order extrapolation mixed finite element matrix model, where the central processing unit time is greatly reduced and the accuracy is maintained.

Topology optimization of Stokes flow with traction boundary conditions using low-order finite elements

We consider topology optimization of Stokes flow with traction boundary conditions using finite elements with low-order velocity-approximation and an element-wise constant hydrostatic pressure. The finite element formulation is stabilized using a penalty on the jump in pressure between adjacent elements. Convergence of solutions to the finite element-discretized topology optimization problem is shown, and several optimization problems are solved using a preconditioned conjugate gradient solver for the finite element matrix problem. Stable convergence to high-quality designs without an excessive number of linear solver iterations is observed, and it is seen that the finite element formulation is not particularly sensitive to the choice of the pressure jump penalty parameter, thus making it a practically useful method.

Theoretical study of the measurement of electric field strength based on the pendular spectra of linear (HCCCN)n (n = 1−3) molecules

The pendular spectra of linear ( H C C C N ) n ( n = 1 − 3 ) molecules at several rotational temperatures under a certain range of electric field strength E are calculated using the finite element matrix diagonalization method. In addition, the normalized “ Q -branch” intensity ( I Q ) and the gradient ( K E ) of I Q under corresponding conditions are investigated. Based on the results, the range and sensitivity of measuring the electric field strength with linear ( H C C C N ) n ( n = 1 − 3 ) molecules are analyzed. A scheme to measure electric field strength based on ( H C C C N ) n ( n = 1 − 3 ) with high sensitivity is proposed and the spatial resolution is also discussed. The feasibility of measuring the electric field strength of the electrostatic Stark decelerator based on a pendular spectrum is studied.

Thickness-resonance waves in underlays of floating screed

The prediction of the reduction of impact sound pressure level ΔL according to annex C of the standard ISO 12354-2 gives an acceptable estimation of the floating floor's performance for thin resilient layers. However, the performance is often largely overestimated for thick resilient layers or for resilient layers combined with thermal layers. One reason for this is that the simplified model doesn't account for the thickness resonances in the underlays which can greatly affect ΔL. This is confirmed by comparing finite element and transfer matrix method simulations with experimental results. This paper establishes the mechanisms leading to the development of these resonance waves and provides some guidelines to estimate their negative effects on the ΔL.

Diffuse sound absorption modelling of complex finite absorbers using a hybrid deterministic-statistical energy analysis approach

This contribution presents a numerical approach to quantify the response of an absorber in a diffuse reverberation room. Conventionally, this is done by considering an infinite absorber coupled to an acoustic halfspace. It is, however, well known that the diffuse absorption coefficient for a finite absorber can be quite different due to what is referred to in literature as the edge effect. A finite size correction has been developed previously, but it is only applicable to homogeneous absorbers and is based on a computationally costly quintuple integration. This contribution presents an alternative approach in which a deterministic model, e.g. using the finite element or modal transfer matrix method, is coupled with a statistical model of the room using a hybrid deterministic-statistical energy analysis framework. With this framework, also the theoretical uncertainty on this diffuse sound absorption that is inherent in the diffuse field assumption can be quantified, i.e. the variance of sound absorption results that can be theoretically expected across an ensemble of reverberation rooms of the same volume. The methodology is numerically and experimentally validated for several absorber types.

Matrix-Free Edge-Domain Decomposition Method for Massively Parallel 3-D Finite Element Simulation With Field-Circuit Coupling

In this article, a novel edge-domain decomposition (EDD) method is proposed to solve 3-D nonlinear finite element (FE) problems of electromagnetic devices and transient field circuit co-simulation. The method applies reduced magnetic vector potential formulation to discretize the physical problem based on 3-D edge elements, and the solution region is divided into many sub-domains that only contain one edge unknown. The solution of lightweight nonlinear sub-domain systems can be massively parallelized, and the neighbor-to-neighbor communication scheme eliminates the need to assemble the global FE matrix. This article also introduces an indirect coupling scheme to handle large eddy currents to interface the EDD FE system with external circuits. The abovementioned algorithms are then implemented on a many-core GPU for transient field circuit co-simulation. The result shows an auto-gauging property, and the comparison with a commercial FE software indicates a speedup of over 43 times with relative error less than 2%.

Influence of Structural Configurations on the Shear Fatigue Damage of the Blade Trailing-Edge Adhesive Joint

Wind turbines are under continuous development for large-scale deployment and oceanization, leading to the requirement of longer blades. The economic losses caused by blade replacement and shutdown have increased. The downtime caused by blade issues in a wind turbine is 8–20% of the total downtime. Many of these blade issues originate from the cracking of the blade trailing edge. The edge is more susceptible to damage due to the complex geometry, manufacturing technique, and operation conditions. The traditional design method and the expensive experimental research are not suitable for the accurate damage analysis of the trailing-edge adhesive because of simplifying assumptions and costs. This study aimed to investigate the influence of trailing-edge structural configurations on the shear fatigue life of the trailing-edge adhesive joint using finite element and stress transformation matrix (STM) methods. The structural configurations of the blade trailing edge included the position of unidirectional fiber layer (UD), chamfer of bonding line, prefabricated components, and outer over-lamination of the trailing edge. In this study, the finite element method was used to simulate the blade structure. The shell element was used for laminates, and the solid element was used for the trailing-edge adhesive joint. The basic shear fatigue properties of the adhesive were obtained by standard component tests. The shear fatigue life of the blade trailing-edge adhesive joint under given load conditions was calculated using the fatigue properties of the adhesive and STM method. The results showed that the angle of chamfering, location of UD, rigidity of the preform, and outer over-lamination all had an obvious influence on the fatigue damage of trailing-edge adhesive. The findings of this study can be used to guide blade structure design and blade production and maintenance.