Three-dimensional Solid Mechanics Research Articles

Leveraging graph neural networks and neural operator techniques for high-fidelity mesh-based physics simulations

Developing fast and accurate computational models to simulate intricate physical phenomena has been a persistent research challenge. Recent studies have demonstrated remarkable capabilities in predicting various physical outcomes through machine learning-assisted approaches. However, it remains challenging to generalize current methods, usually crafted for a specific problem, to other more complex or broader scenarios. To address this challenge, we developed graph neural network (GNN) models with enhanced generalizability derived from the distinct GNN architecture and neural operator techniques. As a proof of concept, we employ our GNN models to predict finite element (FE) simulation results for three-dimensional solid mechanics problems with varying boundary conditions. Results show that our GNN model achieves accurate and robust performance in predicting the stress and deformation profiles of structures compared with FE simulations. Furthermore, the neural operator embedded GNN approach enables learning and predicting various solid mechanics problems in a generalizable fashion, making it a promising approach for surrogate modeling.

An Element Decomposition Method for Three-Dimensional Solid Mechanics

This paper proposes an element decomposition method (EDM) for elastic-static, free vibration and forced vibration analyses of three-dimensional solid mechanics. The problem domain is first discretized using eight-node hexahedral elements. Then, each hexahedron is further subdivided into a set of sub-tetrahedral cells, and the local strains in each sub-tetrahedron are obtained using linear interpolation functions. For each hexahedron, the strain of the whole element is the weighted average value of the local strains, which means only one integration point is adopted to establish the stiffness matrix. To cure the numerical instability of one-point quadrature and improve the accuracy, a variation gradient item is complemented by variance of the local strains. Numerical examples, including both benchmark and practical engineering cases, demonstrate that the present method possesses the following interesting properties compared with the traditional finite element method using the same mesh discretization (1) super accuracy and faster convergence rate; (2) higher computational efficiency; (3) more immune to mesh distortion.

Analysis of an Elasto-Hydrodynamic Seal by Using the Reynolds Equation

This paper reports numerical studies of an Elasto-Hydrodynamic (EHD) seal, which is being developed for supercritical CO2 (sCO2) turbomachinery applications. Current sCO2 turbomachinery suffers from high leakage rates, which is creating a major roadblock to the full realization of sCO2 power technology. The high leakage rates not only penalize the efficiencies but also create environmental concerns due to greenhouse effects caused by the increased CO2 discharge to the atmosphere. The proposed EHD seal needs to work at elevated pressures (10–35 MPa) and temperatures (350–700 °C) with low leakage and minimal wear. The unique mechanism of the EHD seal provides a self-regulated constriction effect to restrict the flow without substantial material contact, thereby minimizing leakage and wear. This work utilizes a physics-based modeling approach. The flow through the gradually narrowing seal clearance is modeled by the well-known Reynolds equation in EHD lubrication theory, while the deformation of the seal is modeled by using the governing equations of three-dimensional solid mechanics. As for the solution methodology, COMSOL’s Thin-Film Flow and Solid Mechanics modules were employed with their powerful capabilities. The numerical results were presented and discussed. It was observed that the Reynolds equation fully coupled with the surface deformation was able to successfully capture the constriction effect. The maximum and minimum leakages were calculated to be 2.25 g/s and 0.1 g/s at P = 5.5 MPa and P = 11 MPa for the design seal, respectively. It was interesting to observe that the seal leakage followed a quadratic trend with increasing pressure differential, which can become advantageous for high-pressure applications such as sCO2 power generation technology.

A Class of Novel Tetrahedron Elements with Curved Surfaces for Three-Dimensional Solid Mechanics Problems with Curved Boundaries

Linear tetrahedral elements with four nodes (Te4) are currently the simplest and most widely used ones in the finite element (FE) developed for solving three-dimensional (3D) mechanics problems. However, the standard Te4 element cannot be used to simulate accurately the 3D problems with curved boundaries because of the flat surfaces. In this paper, we develop a set of new elements having curved surfaces to properly simulate the curved boundaries. At the same time, additional nodes are put on the curved boundaries to improve the accuracy of the approximation. These novel elements are defined as five-noded, six-noded, and seven-noded tetrahedron elements (Te5, Te6, and Te7) according to the number of the nodes in one element. Based on the Te4 FE mesh, a hybrid mesh can be conveniently built for 3D problems with curved boundaries, in which the standard Te4 elements are used for the interior elements, and Te5, Te6, and Te7 elements are used for the curved boundary elements. Compared with the standard FEM using Te4 elements, our hybrid mesh can significantly improve the accuracy of the solutions at the curved boundaries. Several solid mechanics problems are studied using the hybrid meshes to validate the effectiveness of the present new elements.

Shockwave propagation and pressure in aluminum films induced by hot-electron-blast during femtosecond-pulsed laser irradiation

This paper presents a transient analysis of the wave propagation in an aluminum film caused by a femtosecond laser pulse irradiation on the surface. The hot-electron-blast is induced during interaction between a femtosecond laser pulse and the aluminum film. By employing three-dimensional solid mechanics and TTM-TEF (two-temperature and thermal-elastic-force) model, the dynamic wave propagation and dynamic shock pressure P are studied and compared with constant results obtained by previous methods. The low-reflecting boundary conditions have been used to truncate the computational domain to a reasonable size. The dynamic shock pressure P and wave propagation show dynamic change process inside of aluminum films.

An elasto-mechanical unfeelability cloak made of pentamode metamaterials.

Metamaterial-based cloaks make objects different from their surrounding appear just like their surrounding. To date, cloaking has been demonstrated experimentally in many fields of research, including electrodynamics at microwave frequencies, optics, static electric conduction, acoustics, fluid dynamics, thermodynamics and quasi two-dimensional solid mechanics. However, cloaking in the seemingly simple case of three-dimensional solid mechanics is more demanding. Here, inspired by invisible core-shell nanoparticles in optics, we design an approximate elasto-mechanical core-shell 'unfeelability' cloak based on pentamode metamaterials. The resulting three-dimensional polymer microstructures with macroscopic overall volume are fabricated by rapid dip-in direct laser writing optical lithography. We quasi-statically deform cloak and control samples in the linear regime and map the displacement fields by autocorrelation-based analysis of recorded movies. The measured and the calculated displacement fields show very good cloaking performance. This means that one can elastically hide objects along these lines.

ウェーブレットガラーキン法による三次元構造のアダプティブ解析

ウェーブレットガラーキン法による三次元構造のアダプティブ解析

The dynein-triggered ciliary motion in embryonic nodes: an exploratory study based on computational models.

The cilia, presenting a rotational movement in the embryonic nodes, play a crucial role in the left-right specification during embryogenesis. The characteristic architecture of these cilia is based on a cylindrical arrangement of 9 doublet microtubules and the motion of the cilia is triggered by the dynein motors located between adjacent doublets by converting the chemical energy into mechanical work. Restricted by the inherent difficulties of experiments, the dynein activation patterns in moving cilia cannot be directly observed. Thus, the mechanism of nodal ciliary movement is still unclear. In this study, we present computational models of the nodal ciliary ultrastructure based on tomographic images of the ciliary body. By employing time accurate three-dimensional solid mechanics analysis, we investigate the dynein-triggered sliding between adjacent doublet microtubules and simulate the induced ciliary bending. As an exploratory study, two dynein activation patterns are proposed and their rationality is discussed. The mathematical model presented by this paper provides a platform to investigate various assumptions of dynein activity, facilitating us to propose the most possible dynein activation pattern and therefore improving our understandings regarding the protein-beating problems of cilia.

Two mixed finite element formulations with area bubble functions for tetrahedral elements

We address the solution of geometrically linear elastic problems for three-dimensional solid mechanics using tetrahedral finite elements. Starting from a five field weak formulation involving fields for compatible displacements, incompatible displacements, pressure, enhanced strains and stresses, both, the mixed method of incompatible modes and the mixed method of enhanced strains are considered as special cases. As a key idea, area bubble functions are used for both mixed finite element formulations in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test and unique solution of the discrete equation are verified. We also observe that, contrary to a the standard enhanced assumed strain method applied to triangles and tetrahedra our mixed formulations with pressure as an independent variable are not degenerate. In the representative examples firstly a numerical verification of the patch test is obtained. Two additional examples, Cook’s membrane problem and a plate with a ring hole, illustrate the good performance of the presented approaches in comparison to existing finite element formulations.

Singular boundary elements for three-dimensional elasticity problems

The focus of this paper is a set of semi-discontinuous, traction-singular surface elements introduced to help the rigorous boundary integral analysis of problems in three-dimensional solid mechanics. In contrast to the singular boundary elements developed for linear fracture mechanics where the square-root singularity is of primary interest, traction shape functions featuring the proposed four- and eight-node boundary elements can be used to represent power-type singularities of arbitrary order, such as those arising at non-smooth material boundaries and interfaces. Apart from being capable of rigorously handling traction singularities and discontinuities across the domain boundaries and interfaces, these elements also permit a smooth transition to adjacent regular elements. Complemented with a family of suitable displacement and geometry shape functions, the singular surface elements are incorporated into a regularized boundary integral equation method and shown, through a set of benchmark results, to perform well for both static and dynamic problems.

A mixed-enhanced formulation tetrahedral finite elements

This paper considers the solution of problems in three-dimensional solid mechanics using tetrahedral finite elements. A formulation based on a mixed-enhanced treatment involving displacement, pressure and volume effects is presented. The displacement and pressure are used as nodal quantities while volume effects and enhanced modes belong to individual elements. Both small and finite deformation problems are addressed and sample solutions are given to illustrate the performance of the formulation. Copyright © 2000 John Wiley & Sons, Ltd.

A study of implementation schemes for vectorized sparse EBE matrix-vector multiplication

A study of implementation schemes for vectorized sparse element-by-element (EBE) matrix-vector multiplication used in the iterative equation solving of finite element problems is presented. The Hughes, Ferencz and Hallquist scheme and a variation are compared with Torigaki's approach. Computer timing and required memory for each scheme on the IBM 3090/VF vector computer regarding the analysis of large scale three-dimensional solid mechanics problems are presented and results compared.