Finite Elements in Analysis and Design最新文献_第7页

A pure-Lagrangian finite element approach for solving thermo-electrical-mechanical models. Application to electric upsetting 求解热电-力学模型的纯拉格朗日有限元方法。电镦粗的应用

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-28 DOI: 10.1016/j.finel.2025.104433

M. Benítez , A. Bermúdez , P. Fontán , I. Martínez , P. Salgado

In this paper, we introduce a novel numerical procedure for solving fully coupled thermo-electrical-mechanical problems using implicit Runge–Kutta time integration within a purely Lagrangian finite element framework. Our formulation, grounded in continuum mechanics, accurately captures the interdependence of mechanical, thermal, and electrical effects under large deformations. It features a fully coupled thermo-electrical-mechanical Lagrangian model with an elasto-viscoplastic constitutive law, considers six primary variables –velocity, temperature, electric potential, plastic deformation gradient, an internal strain hardening variable, and a Lagrange multiplier for enforcing contact conditions– and employs a pure-Lagrangian description. This ensures the computational domain remains fixed and known a priori, simplifies the tracking of free surfaces, and eliminates convective terms. To validate our approach, we solve several axisymmetric benchmark problems and analyze convergence rates in both time and space. Moreover, our numerical results show excellent agreement with the solution obtained using commercial packages for an in-die electric upsetting process.

本文在纯拉格朗日有限元框架下，利用隐式龙格-库塔时间积分，提出了求解热电-机械全耦合问题的一种新的数值方法。我们的配方以连续介质力学为基础，准确地捕捉了大变形下机械、热和电效应的相互依存关系。它具有具有弹粘塑性本构律的完全耦合热电机械拉格朗日模型，考虑了六个主要变量-速度，温度，电势，塑性变形梯度，内部应变硬化变量和用于强制接触条件的拉格朗日乘数-并采用纯拉格朗日描述。这确保了计算域保持固定和先验已知，简化了自由曲面的跟踪，并消除了对流项。为了验证我们的方法，我们解决了几个轴对称基准问题，并分析了时间和空间上的收敛速度。此外，我们的数值计算结果与在模内电镦过程中使用商业封装得到的解非常吻合。

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引用次数: 0

Efficient co-rotational formulation for 3D composite beams with two-directional interlayer slip 具有双向层间滑移的三维组合梁的有效共转公式

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-26 DOI: 10.1016/j.finel.2025.104432

Yassir Wardi, Pisey Keo, Mohammed Hjiaj

In this paper, we present a novel 3D nonlinear formulation for two-layered composite beams that accounts for interlayer slip in both longitudinal and lateral directions. Warping effects are included in a simplified manner, assuming that the warping of each layer does not contribute to the stress resultants of each section, allowing the use of the classical St. Venant warping function to define the warping shape of each subsection. The second-order approximation of the Green–Lagrange strain tensor, combined with linear constitutive laws, is integrated into the principle of virtual work to derive the tangent stiffness matrix of the composite element and its corresponding internal force. To address membrane and slip locking issues, we propose a new averaging strain technique, complemented by quadratic interpolation functions for the axial displacement of the two layers. To account for large displacements and rotations, the co-rotational approach is adopted. The co-rotated local reference frame is constructed by connecting end nodes located at the shear center of the bottom layer of the composite beam. As a result, special treatments are employed to address eccentric forces applied to the top layer of the composite beam. Finally, the performance of the proposed formulation is evaluated using four representative examples.

在本文中，我们提出了一个新的三维非线性公式的两层组合梁，考虑层间滑移在纵向和横向方向。假设每一层的翘曲不影响每个部分的应力结果，以简化的方式包括翘曲效果，允许使用经典的St. Venant翘曲函数来定义每个分段的翘曲形状。将格林-拉格朗日应变张量的二阶近似，结合线性本构定律，与虚功原理相结合，导出复合单元的切向刚度矩阵及其对应的内力。为了解决膜和滑移锁紧问题，我们提出了一种新的平均应变技术，并辅以两层轴向位移的二次插值函数。为了考虑较大的位移和旋转，采用了共旋转方法。通过连接位于组合梁底层剪切中心的端节点来构建共旋转局部参考框架。因此，采用特殊处理来解决施加在复合梁顶层的偏心力。最后，用四个代表性的例子对所提公式的性能进行了评价。

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引用次数: 0

An experimental and numerical dynamic study of thick sandwich beams using a mixed {3,2}-RZT formulation 使用{3,2}-RZT混合公式的厚夹层梁的实验和数值动力学研究

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-22 DOI: 10.1016/j.finel.2025.104435

Matteo Sorrenti, Marco Gherlone

<div><div>This work presents some numerical and experimental validations of the free-vibration behaviour of thick sandwich beams using the mixed {3,2}-Refined Zigzag Theory (<span><math><mrow><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span>). The <span><math><mrow><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span> formulation enhances the Timoshenko's kinematics with a piece-wise zigzag cubic distribution of the axial displacement, and a smoothed parabolic variation for the transverse deflection. Simultaneously, an a-priori assumption is made for the transverse normal stress and the transverse shear one: the former is assumed to be a third-order power series expansion of the thickness coordinate, while the latter is derived through the integration of Cauchy's equations. The equations of motion and consistent boundary conditions for the free-vibration problem are derived through the Hellinger-Reissner (HR) theorem. Taking advantage of the C<sup>0</sup>-continuity requirement in the mixed governing functional, a simple two-node beam finite element (FE) is formulated, i.e., the <span><math><mrow><mn>2</mn><mi>B</mi><mo>−</mo><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span> element. The analytical and FE performances of the proposed <span><math><mrow><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span> model are first addressed by means of a comparison with high-fidelity 3D FE models. Subsequently, an experimental campaign is conducted using LASER Doppler Vibrometry (LDV) to evaluate the modal parameters of a series of thick sandwich beams made of aluminium alloy face-sheets and Rohacell® WF110 core. The experimental results concerning the natural frequencies and modal shapes of the thick sandwich beam specimens under free-free boundary conditions are compared with those given by <span><math><mrow><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span> and high-fidelity 3D FE models. The numerical-experimental assessment highlights the effect of core and face-sheet thickness on frequency estimations, as well as the complexity of reproducing in the numerical model the experimental uncertainties. In general, the <span><math><mrow><mn>2</mn><mi>B</mi><mo>−</mo><msubsup><mtext>RZT</mtext><mrow><mo>{</mo><mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow><mo>}</mo></mrow><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow></msubsup></mrow></math></span

本文采用{3,2}-精炼之字形理论（RZT{3,2}(m)）对厚夹层梁的自由振动特性进行了一些数值和实验验证。RZT{3,2}(m)公式通过轴向位移的分段之形三次分布和横向挠度的平滑抛物线变化增强了Timoshenko的运动学。同时，对横向正应力和横向剪应力进行了先验假设，其中横向正应力为厚度坐标的三阶幂级数展开式，横向剪应力为柯西方程的积分式。利用Hellinger-Reissner （HR）定理导出了自由振动问题的运动方程和一致边界条件。利用混合控制泛函中的c0 -连续性要求，建立了简单的两节点梁有限元（FE），即2B−RZT{3,2}(m)单元。首先通过与高保真三维有限元模型的比较，讨论了所提出的RZT{3,2}(m)模型的分析性能和有限元性能。随后，使用激光多普勒振动仪（LDV）进行了一项实验活动，以评估由铝合金面板和Rohacell®WF110芯制成的一系列厚夹层梁的模态参数。将自由-自由边界条件下厚夹层梁试件固有频率和模态振型的实验结果与RZT{3,2}(m)和高保真三维有限元模型给出的结果进行了比较。数值-实验评估强调了岩心和面板厚度对频率估计的影响，以及在数值模型中再现实验不确定性的复杂性。总的来说，2B−RZT{3,2}(m)单元公式在厚夹层梁动力分析中显示出其准确性和计算优势。

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引用次数: 0

Integration of hierarchical quadrature element method with a minimum-increment remeshing strategy for simulating coupled thermo-mechanical fracture in quasi-brittle materials 准脆性材料热-力耦合断裂模拟的分层正交元法与最小增量重网格策略集成

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-18 DOI: 10.1016/j.finel.2025.104434

Sihua Hu , Xing Luo , Wei Xiang

This paper presents a p-version finite element framework for analyzing the thermal fracture behavior of quasi-brittle materials under coupled thermo-mechanical loadings. The proposed formulation, based on the hierarchical quadrature element method (HQEM), enables accurate capture of temperature gradients even on relatively coarse meshes. Its accuracy in simulating heat conduction and thermally induced deformation is validated against ABAQUS results.

The HQEM is integrated with the virtual crack closure method to compute fracture parameters under combined thermal and mechanical loadings, significantly reducing mesh refinement and preprocessing effort compared to conventional h-version FEM. To efficiently track complex crack paths, a minimum-increment remeshing strategy is introduced, which controls element growth while preserving the geometric accuracy of crack paths during iterative crack propagation analysis, significantly reducing the computational cost associated with frequent remeshing. Applications to four representative numerical examples demonstrate excellent agreement with existing literature, confirming the reliability and accuracy of the proposed approach for coupled thermo-mechanical fracture analysis.

本文提出了一种用于分析准脆性材料在热-力耦合载荷作用下热断裂行为的p型有限元框架。提出的公式，基于分层正交单元法（HQEM），即使在相对粗糙的网格上也能准确捕获温度梯度。与ABAQUS模拟结果对比，验证了其模拟热传导和热致变形的准确性。HQEM与虚拟裂纹闭合方法相结合，可以计算热和机械联合载荷下的断裂参数，与传统的h型有限元法相比，大大减少了网格细化和预处理工作量。为了有效地跟踪复杂裂纹路径，引入了最小增量重网格策略，该策略在控制单元增长的同时，在迭代裂纹扩展分析过程中保持了裂纹路径的几何精度，显著降低了频繁重网格的计算成本。通过对四个典型数值算例的分析，验证了本文提出的热-力耦合断裂分析方法的可靠性和准确性。

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引用次数: 0

Two stabilized finite element methods based on local polynomial pressure projection for the steady-state Navier–Stokes–Darcy problem 求解稳态Navier-Stokes-Darcy问题的两种基于局部多项式压力投影的稳定有限元方法

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-15 DOI: 10.1016/j.finel.2025.104420

Liyun Zuo , Guangzhi Du

This study presents two stabilized finite element methods based on local polynomial pressure projections for the mixed steady-state Navier–Stokes–Darcy problem by utilizing the equal order finite element pairs, the

P_{1}

-

P_{1}

-

P_{1}

and

P_{2}

-

P_{2}

-

P_{2}

element pairs, for approximating the fluid velocity, kinematic pressure and dynamic pressure, respectively. The presented stabilized methods possess many chief characteristics, for instance, parameter free, simple calculation, element level implementation. The optimal error estimates are established. Finally, some comprehensively numerical tests are reported to examine the efficiency and robustness of the proposed algorithms.

针对混合稳态Navier-Stokes-Darcy问题，利用等阶有限元对P1-P1-P1和P2-P2-P2单元对分别逼近流体速度、运动压力和动压力，提出了两种基于局部多项式压力投影的稳定有限元方法。所提出的稳定方法具有无参数、计算简单、单元级实现等主要特点。建立了最优误差估计。最后，通过数值实验验证了所提算法的有效性和鲁棒性。

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引用次数: 0

A Green’s function driven mesh reduction technique for obtaining closed-form solutions of uniform Euler–Bernoulli beams on two-parameter elastic foundations 双参数弹性基础上均匀欧拉-伯努利梁闭型解的格林函数驱动网格化简方法

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-14 DOI: 10.1016/j.finel.2025.104418

Juan Camilo Molina-Villegas, Julián Esteban Ossa Gómez

This paper presents the formulation of the Green’s Function Stiffness Method (GFSM) for the static analysis of linearly elastic uniform Euler–Bernoulli beams on two-parameter elastic foundations subjected to arbitrary external loads. The GFSM is a mesh-reduction method closely related to the Finite Element Method (FEM) family, offering a means to compute closed-form solutions for framed structures. It is based on a strong-form formulation and decomposes the element-level response into homogeneous and fixed (particular) components, the latter obtained analytically using Green’s functions of fixed-end elements. The method retains essential FEM features — including shape functions, stiffness matrices, and fixed-end force vectors — while extending the capabilities of the Transcendental Finite Element Method (TFEM), a FEM variant that employs exact shape functions. In this context, the GFSM serves as a post-processing enhancement that transforms the approximate TFEM solution into an exact closed-form. A defining characteristic of the GFSM is that its formulation relies solely on the solution of the homogeneous form of the governing differential equations — specifically, the shape functions and stiffness matrix coefficients that constitute the core of the TFEM. The effectiveness of the GFSM is demonstrated through two examples, where its results are compared against those obtained from TFEM with varying levels of mesh refinement.

本文提出了在任意外荷载作用下双参数弹性基础上线弹性均匀欧拉-伯努利梁静力分析的格林函数刚度法（GFSM）的公式。GFSM是一种与有限元法（FEM）家族密切相关的网格缩减方法，提供了一种计算框架结构封闭形式解的方法。它基于强形式公式，并将单元级响应分解为齐次和固定（特定）分量，后者使用固定端单元的格林函数解析得到。该方法保留了FEM的基本特征，包括形状函数、刚度矩阵和固定端力向量，同时扩展了超越有限元法（TFEM）的功能，TFEM是一种采用精确形状函数的FEM变体。在这种情况下，GFSM作为后处理的增强，将近似的TFEM解转换为精确的封闭形式。GFSM的一个决定性特征是，它的公式完全依赖于控制微分方程的齐次形式的解，特别是构成TFEM核心的形状函数和刚度矩阵系数。通过两个实例证明了GFSM的有效性，并将其结果与不同网格细化水平的TFEM结果进行了比较。

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引用次数: 0

High-resolution thermal simulation framework for extrusion-based additive manufacturing of complex geometries 复杂几何形状挤压增材制造的高分辨率热模拟框架

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-14 DOI: 10.1016/j.finel.2025.104410

Dhruv Gamdha, Kumar Saurabh, Baskar Ganapathysubramanian, Adarsh Krishnamurthy

Accurate simulation of the printing process is essential for improving print quality, reducing waste, and optimizing the printing parameters of extrusion-based additive manufacturing. Traditional additive manufacturing simulations are very compute-intensive and are not scalable to simulate even moderately sized geometries. In this paper, we propose a general framework for creating a digital twin of the dynamic printing process by performing physics simulations with the intermediate print geometries. Our framework takes a general extrusion-based additive manufacturing G-code, generates an analysis-suitable voxelized geometry representation from the print schedule, and performs physics-based (transient thermal) simulations of the printing process. Our approach leverages adaptive octree meshes for both geometry representation as well as for fast simulations to address real-time predictions. We demonstrate the effectiveness of our method by simulating the printing of complex geometries at high voxel resolutions with both sparse and dense infills. Our results show that this approach scales to high voxel resolutions and can predict the transient heat distribution as the print progresses. Because the simulation runs faster than real print time, the same engine could, in principle, feed thermal predictions back to the machine controller (e.g., to adjust fan speed or extrusion rate). The present study establishes the computational foundations for a real-time digital twin, which can be used for closed control loop control in the future.

打印过程的精确模拟对于提高打印质量、减少浪费和优化基于挤压的增材制造的打印参数至关重要。传统的增材制造模拟是非常计算密集型的，并且不能扩展到模拟中等大小的几何形状。在本文中，我们提出了一个通用框架，通过对中间印刷几何形状进行物理模拟来创建动态印刷过程的数字孪生。我们的框架采用通用的基于挤压的增材制造g代码，从打印计划中生成适合分析的体素化几何表示，并执行基于物理（瞬态热）的打印过程模拟。我们的方法利用自适应八叉树网格进行几何表示以及快速模拟以解决实时预测。我们通过模拟在高体素分辨率下具有稀疏和密集填充的复杂几何图形的打印来证明我们方法的有效性。我们的结果表明，该方法适用于高体素分辨率，并且可以预测打印过程中的瞬态热分布。由于模拟运行速度快于实际打印时间，因此原则上，相同的发动机可以将热预测反馈给机器控制器（例如，调整风扇速度或挤出速率）。本研究奠定了实时数字孪生的计算基础，可用于未来的闭环控制。

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引用次数: 0

A rotation-based geometrically nonlinear spectral Reissner–Mindlin shell element 基于旋转的几何非线性谱Reissner-Mindlin壳单元

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-13 DOI: 10.1016/j.finel.2025.104416

Nima Azizi , Wolfgang Dornisch

In this paper, we propose a geometrically nonlinear spectral shell element based on Reissner–Mindlin kinematics using a rotation-based formulation with additive update of the discrete nodal rotation vector. The formulation is provided in matrix notation in detail. Additionally, we highlight the advantages of the spectral element method (SEM) in combination with Gauss–Lobatto–Legendre quadrature regarding the computational costs to generate the element stiffness matrix. To assess the performance of the new formulation for large deformation analysis, we compare it to three other numerical methods. One of these methods is a non-isoparametric SEM shell using the geometry definition of isogeometric analysis (IGA), while the other two are IGA shell formulations which differ in the rotation interpolation. All formulations base on Rodrigues’ rotation tensor. Through the solution of various challenging numerical examples, it is demonstrated that although IGA benefits from an exact geometric representation, its influence on solution accuracy is less significant than that of shape function characteristics and rotational formulations. Furthermore, we show that the proposed SEM shell, despite its simpler rotational formulation, can produce results comparable to the most accurate and complex version of IGA. Finally, we discuss the optimal SEM strategy, emphasizing the effectiveness of employing coarser meshes with higher-order elements.

本文提出了一种基于Reissner-Mindlin运动学的几何非线性谱壳单元，使用基于旋转的公式，并对离散节点旋转向量进行加性更新。以矩阵的形式给出了详细的公式。此外，我们强调了谱元法（SEM）与Gauss-Lobatto-Legendre正交相结合在生成单元刚度矩阵的计算成本方面的优势。为了评估新公式在大变形分析中的性能，我们将其与其他三种数值方法进行了比较。其中一种方法是采用等几何分析（IGA）的几何定义的非等参数SEM壳，另两种方法是不同旋转插值的IGA壳公式。所有的公式都基于Rodrigues旋转张量。通过各种具有挑战性的数值算例的求解，证明了IGA虽然受益于精确的几何表示，但其对求解精度的影响不如形状函数特征和旋转公式的影响显著。此外，我们表明，尽管所提出的SEM外壳的旋转配方更简单，但可以产生与最精确和最复杂版本的IGA相当的结果。最后，我们讨论了最优的扫描电镜策略，强调了采用高阶元素的粗网格的有效性。

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引用次数: 0

A novel quasi-smooth manifold element method for structural transient heat conduction analysis with radiation and nonlinear boundaries 一种新的具有辐射和非线性边界的结构瞬态热传导分析的准光滑流形元方法

IF 3.5 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-13 DOI: 10.1016/j.finel.2025.104428

Xin Ye , Shanzhi Liu , Weibin Wen , Pan Wang , Jun Liang

This study proposes a novel quasi-smooth manifold element (QSME) method to solve structural heat conduction problem. Compared with the conventional finite element (FE) method, the main advantage of the QSME method is the use of high-order local approximation. This ensures the continuity of first-order derivatives at element nodes, enhancing computation accuracy. The results show that the QSME method has high computation accuracy and efficiency. It can effectively solve the nonlinear thermal radiation problem of complex geometries. Under the same degrees of freedom (DOFs), the QSME method achieves at least one-order magnitude higher accuracy than the conventional FE method. Moreover, compared with the FE method, it attains faster convergence rate and requires far less DOFs to achieve the roughly same solution accuracy. This method provides an efficient computational tool for heat conduction analysis and coupled multi-physics simulations.

提出了一种求解结构热传导问题的准光滑流形元（QSME）方法。与传统有限元方法相比，QSME方法的主要优点是采用了高阶局部逼近。这保证了单元节点上一阶导数的连续性，提高了计算精度。结果表明，该方法具有较高的计算精度和效率。它能有效地解决复杂几何形状的非线性热辐射问题。在相同的自由度下，QSME方法的精度比传统有限元方法提高了至少一个数量级。此外，与有限元方法相比，它具有更快的收敛速度和更少的自由度以达到大致相同的解精度。该方法为热传导分析和多物理场耦合模拟提供了有效的计算工具。

引用次数: 0

A computationally efficient hybrid technique for analyzing three-dimensional effects in contacts 一种计算效率高的分析接触面三维效应的混合技术

IF 3.1 3区工程技术 Q1 MATHEMATICS, APPLIED

Finite Elements in Analysis and Design

Pub Date : 2025-08-08 DOI: 10.1016/j.finel.2025.104417

P. Pradhan, H. Murthy

The accuracy of FE analysis depends on the element size and integration technique used and requires significant computational effort for 3D contact problems involving large stress gradients. Therefore, contacts with similar geometries in the third dimension are typically analyzed using 2D techniques. Analysis of such 2D contacts using infinite series to solve the governing singular integral equations requires much lesser computation effort than even 2D FE analysis. However, it neglects the effect of finite dimension in the third direction due to which the contact is not under plane conditions. To investigate the effect of finiteness of third dimension in a computationally efficient manner, a hybrid technique is developed for 3D contact analysis that inherits the versatility of FE analysis and the computational efficiency of the series solution. Its results are compared to those of a detailed 3D FE analysis with fine mesh and full integration to ascertain its efficacy. They match very well in most of the contact regions except for a small difference in peak pressure near the free edge of contact.

有限元分析的准确性取决于所使用的单元尺寸和集成技术，并且对于涉及大应力梯度的三维接触问题需要大量的计算工作。因此，在三维空间中具有相似几何形状的接触通常使用二维技术进行分析。利用无穷级数求解控制奇异积分方程对这种二维接触进行分析所需的计算量比二维有限元分析要少得多。然而，它忽略了在非平面条件下接触的第三方向上有限尺寸的影响。为了有效地研究三维有限性对三维接触分析的影响，在继承有限元分析的通用性和级数解的计算效率的基础上，提出了一种三维接触分析的混合方法。将其结果与精细网格和完全集成的详细三维有限元分析结果进行比较，以确定其有效性。它们在大多数接触区域非常匹配，除了在接触自由边缘附近的峰值压力有很小的差异。

引用次数: 0