International Journal of Computational Methods

Recent breakthroughs in deep-learning algorithms enable dreams of artificial intelligence (AI) getting close to reality. AI-based technologies are now being developed rapidly, including service and industrial robots, autonomous and self-driving vehicles. This work proposes Two-Way Deepnets (TW-Deepnets) trained using the physics-law-based models such as finite element method (FEM), smoothed FEM (S-FEM), and meshfree models, for real-time computations of both forward and inverse mechanics problems of materials and structures. First, unique features of physics-law-based models and data-based models are analyzed in theory. The training characteristics of deepnets for forward problems governed by physics-laws are then investigated, when an FEM (or S-FEM) model is used as the trainer. The training convergence rates of such an FEM-AI model are examined in relation to the property of the system matrix of the FEM model for deepnets. Next, a study on the training characteristics of deepnets for inverse problems, when the forward FEM-trained AI Deepnets are used as the trainer to train an AI model for inverse analyses. Next, a discussion is conducted on the roles of regularization techniques to overcome the ill-posedness of inverse problems in deepnet structures for noisy data. Finally, TW-Deepnets (FEM-AI and AI-AI models) are presented for real-time analyses of both forward and inverse problems of materials and structures with high-dimensional parameter space. The major finding of this study is as follows: (1) The understandings on the fundamental features of both data-based and physics-based methods is critical for creations of novel game-changing computational methods, which take advantages of both types of methods; (2) The good property of the system matrix of FEM allows effective training of FEM-AI deepnets for forward mechanics problems; (3) Our new technique to training inverse deepnets using FEM-AI deepnets as a surrogate model offers an innovative means, to effectively train deepnets for solving inverse mechanics problems; (4) The TW-Deepnets is capable of performing real-time analysis of both forward and inverse problems of materials and structures with high-dimensional parameter spaces; (5) Such TW-Deepnets can be easily utilized by the mass: a transformative new concept of AI-enabling democratization of complicated computational technology in modeling and simulation.

In this study, a hybrid method for density-related topology optimization is proposed, which consists of two parts: the discrete level-set method (LSM) based on solid isotropic material with penalization (SIMP) and the structural boundary extraction method based on support vector machine (SVM). In the hybrid method, the SIMP method is implemented to create new holes which are inserted into the topological structure obtained by the discrete LSM. SVM is utilized for extracting the boundary of the structure obtained by the SIMP-based discrete LSM. Based on the clear boundary extracted by SVM, a smooth boundary can be further obtained after data filtering. Four numerical examples are used to test the advantages of the hybrid method.

In order to optimize the geometric parameters of cutting edge for the finishing machining of 30Cr alloy steel, a two-dimensional (2D) finite element (FE) model of orthogonal cutting was built with FE software AdvantEdge. The optimized methodology of the cutting edge geometric parameters was likewise proposed based on the simulated results. Thereafter, the geometric parameters of the cutting edge were optimized based on a comprehensive criterion that combines chip deformation coefficient and tool stress. The chip deformation coefficient indirectly determines the surface roughness, whereas tool stress determines tool wear, thereby affecting the dimensional precision of the components. The rake angle ranges from 12${}^{\circ }$ to 20${}^{\circ }$, while the cutting edge radius ranges from 12$\mu$m to 20$\mu$m in the optimization process. The optimal rake angle for the finishing machining 30Cr alloy steel is 16${}^{\circ }$, while the optimal cutting edge radius is 14$\mu$m with a given relief angle of 7${}^{\circ }$.

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

Few inverted or poor quality elements can ruin both the accuracy and efficiency of numerical solutions. This paper proposes improved mesh untangling and quality improvement methods using the modified mesh quality metric for 2D triangular meshes. The new mesh quality metric ensures highly inverted elements assign higher penalty when the signed area of the element is negative, and also produces more accurate signed areas for valid elements. Therefore, the proposed methods can quickly remove inverted elements and also improve worst element qualities compared with the existing methods. We also combine Delaunay-based edge flip with optimization-based mesh untangling and quality improvement methods to further improve worst element qualities. Numerical results show that the proposed methods yield better quality meshes for improvement of the worst quality and are faster than existing methods.

In this paper, we study a local convergence analysis of a family of iterative methods with sixth and seventh order convergence for nonlinear equations, which was established by [Cordero et al. [2010] in “A family of iterative methods with sixth and seventh order convergence for nonlinear equations,” Math. Comput. Model. 52, 1190–1496]. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the sixth derivative. In our work, we make an attempt to study and establish a local convergence theorem by using only hypotheses the first derivative of the function and Lipschitz constants. We can also obtain error bounds and radii of convergence based on our results. Hence, the applicability of the methods is expanded. Moreover, we consider some different numerical examples and obtain the radii of convergence centered at the solution for different parameter values $𝜃$ of the family. Furthermore, the basins of attraction of the family with different parameter values are also studied, which allow us to distinguish between the good and bad members of the family in terms of convergence and stable properties, and help us find the members with better or the best stable behavior.

A stabilization method of the F-bar aided edge-based smoothed finite element method with four-node tetrahedral elements (F-barES-FEM-T4), which is named SymF-barES-FEM-T4, is proposed. The proposed formulation aims to remove the instability associated with energy divergence of F-barES-FEM-T4 in dynamic analysis. The stiffness matrix of F-barES-FEM-T4 is asymmetric due to the adoption of F-bar method, meanwhile that of SymF-barES-FEM-T4 is symmetrized so that the dynamics system of the formulation is stabilized. The differences between F-barES-FEM-T4 and SymF-barES-FEM-T4 in their formulations are only the definition of the $B$-matrix and the corresponding volume in the internal force calculation. Therefore, SymF-barES-FEM-T4 is applicable in dynamic explicit schemes as is the case with F-barES-FEM-T4. A few demonstrations of dynamic explicit analysis for nearly incompressible materials reveal that SymF-barES-FEM-T4 conserves the total energy and does not cause energy divergence in the free vibration problems. They also reveal that SymF-barES-FEM-T4 has a moderate capability in the suppression of pressure checkerboarding: worse than F-barES-FEM-T4 but better than Selective ES/NS-FEM-T4.

In the present study, a three-dimensional discontinuous deformation analysis (3D DDA) method was developed and used to investigate the effect of slope geometrical characteristics on rockfall behaviors. The slope geometrical characteristics of rockfall, such as the slope height, slope angle and slope shape, were summarized, and the corresponding calculation models were proposed. The algebraic computation and contact mechanics of the three-dimensional (3D) global contact theory were illustrated and implemented into the 3D DDA method. The accuracy of the developed 3D DDA method was verified through the four basic movement forms of rockfall: freefalling (oblique projectile), sliding, rolling and bouncing. Numerical examples were performed to study rockfall behaviors under different slope geometrical characteristics by analyzing the kinetic energy or velocity conversion, motion trajectories, and damage degree and range of moving blocks. The results demonstrate that the 3D DDA method can accurately simulate rockfall movement, which is significantly affected by slope geometrical characteristics. Furthermore, some suggestions for slope excavation and protection were put forward.

Cell- and node-based smoothing MITC3-finite elements for static and free vibration analysis of laminated composite and functionally graded plates are proposed in this paper. It is based on a higher-order shear deformation theory which accounts for a higher-order variation of in-plane displacements and traction-free boundary conditions on top and bottom surfaces of transverse shear stresses. Novel CS-MITC3 and NS-MITC3 finite elements are proposed in which an MITC3 three-node triangle element with seven degrees-of-freedom per nodes that only requires the ${\mathrm{\text{C}}}^0$-type continuity is developed, and then cell- and node-based smoothing techniques are applied. Numerical results for the laminated composite and functionally graded plates are proposed to validate the accuracy of developed theory and to investigate the effects of material distribution and side-to-thickness ratio on the deflection, stresses and natural frequencies of the plates.

The paper proposes a computational approach to simulate the dynamic responses of composite plate resting on a Pasternak foundation subjected a moving load using the moving element method (MEM). The plate element stiffness matrix is formulated in a coordinate system which moves with the load. The main convenience is that the load is static in this coordinate system, which avoids the updating of the load locations due to the change of the contact points with the elements. The effects of the Pasternak foundation, energy dissipation mechanisms, load’s velocity, material properties on the dynamic responses of the composite plates are investigated.

Based on the crash test of new energy vehicles, the mechanical response data of power batteries during the collision process were collected, and the average impact strength curve of power batteries of typical new energy vehicles in China was obtained. The average impact strength curve was mathematically processed to obtain the impact strength characteristic value and tolerance by using the equivalent trapezoidal wave and the least square method, thereby determining the test conditions of the dynamic strength of the domestic new energy vehicle power battery. The differences are analyzed by comparing with ISO 12405-3 test conditions, which provides an important reference for the revision of power battery test standards in the future.

Repair is now increasingly used for damaged structures due to its distinguished advantages in prolonging the service life of structures, substantial saving in materials, and preservation of related crucial resources. This work presents a novel filling-in repair procedure for localized damages in panel structures, and conducts a series of theory and numerical studies to examine the repair effects in terms of stress concentration. It aims providing some fundamental understandings important in the design of repairing procedures for effective filling-in repairs to mitigate stress concentrations. First, the damaged zone (or cracks) is “cut-off” by simply drilling a hole through the panel to remove possible stress singularity. The hole is then filled-in with a circular patch of slightly smaller diameter made of similar material of the panel using adhesives. Stress concentration is then investigated to reveal the possible ranges of the stress concentration factors (SCF), through both analytical and numerical means using finite element (FE) models. The repair effects are then examined for repaired plates with filled-in holes in terms of SCF reduction, comparing with plates with open-holes. Both aluminum and composite plates are studied, and the filling-in repairs are carried out using epoxy adhesives. It is found that the SCF increases with the adhesive thickness, decreases with hole radius and Young’s modulus of adhesives. Parametric studies are performed to produce two-stage curve-fitted formulae for SCF predictions from given hole radius, physical properties and thickness of the adhesive layer, which are useful for the design and assessment of the repairs, leading to a prolonged life of services of structures.

The effects of $T$-stress on the stress field, creep zone and constraint effect of the mode I crack tip in power-law creeping solids are presented based on finite element (FE) analysis in the paper. The characteristics of the crack tip field in power-law creep solids by considering low negative $T$-stress and high positive $T$-stress are studied in the paper. The differences of $T$-stress effect on the crack tip field between power-law creeping solids and elastoplastic materials are also clarified. A modified parameter is proposed to characterize the influence of $T$-stress on creep zone. The constraint parameter $Q$ under both small-scale creep and large-scale creep with various $T$-stresses for the modified boundary layer (MBL) model and various specimens with different crack depths are given. The applicability and the limitation of the MBL model for creep crack are also investigated. The inherent connection between $T$-stress and $Q$-parameter is discussed. The investigations given in this paper can further promote the understanding of $T$-stress effect and constraint effect on the mode I creep crack.

To improve the safety performance and maintain light weight of composite automotive bumper beams subjected to low-velocity impacts, a structural optimization and a novel reliability analysis procedure for multi-objectives are established in this paper. Both longitudinal and corner pendulum impacts are considered, and the optimized bumper beam has significant improvements. Compared with the original composite bumper beam, the section force of the crash box in the corner pendulum impact and the mass of the optimal bumper beam are decreased by 9.6% and 20.3%, respectively. The novel reliability analysis results show that the knee point may be less reliable whereas the design with inferior knee point is of higher reliability. Therefore, a design on an inferior knee point may be more attractive in engineering design practices. Our numerical results have demonstrated that the optimization procedure established in this paper to improve the performance of composite bumper beams and our proposed reliability analysis method considering multi-objectives are effective and useful in the engineering design of bumper beams. These techniques are also useful for the design of other types of structural components where weight and safety are of importance.

In this work, a new approach for aerodynamic optimization of horizontal axis wind turbine (HAWT) airfoil is presented. This technique combines commercial computational fluid dynamics (CFD) codes with differential evolution (DE), a reliable gradient-free global optimization method. During the optimization process, commercial CFD codes are used to evaluate aerodynamic characteristics of HAWT airfoil and an improved DE algorithm is utilized to find the optimal airfoil design. The objective of this research is to maximize the aerodynamic coefficients of HAWT airfoil at the design angle of attack (AOA) with specific ambient environment. The airfoil shape is modeled by control points which their coordinates are design variables. The reliability of CFD codes is validated by comparing the analytical results of a typical HAWT airfoil with its experimental data. Finally, the optimal design of wind turbine airfoil is evaluated about aerodynamic performance in comparison with existing airfoils and some discussions are performed.