On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra

In this paper, we develop the $$A_\infty $$ A ∞ -analog of the Maurer-Cartan simplicial set associated to an $$L_\infty $$ L ∞ -algebra and show how we can use this to study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of $$A_\infty $$ A ∞ -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) $$A_\infty $$ A ∞ -algebras to simplicial sets, which sends a complete curved $$A_\infty $$ A ∞ -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads.

Large Deformation Problem of Bimodular Functionally-Graded Thin Circular Plates Subjected to Transversely Uniformly-Distributed Load: Perturbation Solution without Small-Rotation-Angle Assumption

In this study, the large deformation problem of a functionally-graded thin circular plate subjected to transversely uniformly-distributed load and with different moduli in tension and compression (bimodular property) is theoretically analyzed, in which the small-rotation-angle assumption, commonly used in the classical Föppl–von Kármán equations of large deflection problems, is abandoned. First, based on the mechanical model on the neutral layer, the bimodular functionally-graded property of materials is modeled as two different exponential functions in the tensile and compressive zones. Thus, the governing equations of the large deformation problem are established and improved, in which the equation of equilibrium is derived without the common small-rotation-angle assumption. Taking the central deflection as a perturbation parameter, the perturbation method is used to solve the governing equations, thus the perturbation solutions of deflection and stress are obtained under different boundary constraints and the regression of the solution is satisfied. Results indicate that the perturbation solutions presented in this study have higher computational accuracy in comparison with the existing perturbation solutions with small-rotation-angle assumption. Specially, the computational accuracies of external load and yield stress are improved by 17.22% and 28.79% at most, respectively, by the numerical examples. In addition, the small-rotation-angle assumption has a great influence on the yield stress at the center of the bimodular functionally-graded circular plate.

Improvement of the Zienkiewicz–Zhu Error Recovery Technique Using a Patch Configuration

The Zienkiewicz–Zhu (ZZ) super-convergent patch recovery technique based on a node neighborhood patch configuration is used most widely for recovery of the stress field of a finite element analysis. In this study, an improved ZZ recovery technique using element neighborhood patch configuration is proposed. The improved recovery procedure is based on recovery of the stress field in the least-squares sense over an element patch that consists of the union of the elements surrounding the element under consideration. The proposed patch configuration provides more sampling points and improves the performance of the standard ZZ recovery technique. The effectiveness and reliability of the improved ZZ recovery approach is demonstrated through plane elastic and plastic plate problems. The problem domain is discretized with triangular and quadrilateral elements of different sizes. A comparison of the quality of error estimation using the ZZ recovery of derivative field and recovery of the displacement field using similar element neighborhood patch configurations is also presented. The numerical results show that the ZZ recovery technique and the displacement recovery technique, using a modified patch configuration, yield better results, convergence rate, and effectivity as compared with the standard ZZ super-convergent patch recovery technique. It is concluded that the improved ZZ recovery technique-based adaptive finite element analysis is very effective for converging a predefined accuracy with a significantly smaller number of degrees of freedom, especially in an elastic problem. It is also concluded that the improved ZZ recovery technique captures the plastic deformation problem solution errors more reliably than the standard ZZ recovery technique.

ACTUATION AND MOTION CONTROL OF FLEXIBLE ROBOTS: SMALL DEFORMATION PROBLEM

This paper introduces a new computational approach for the articulated joint/deformation actuation and motion control of robot manipulators with flexible components. Oscillations due to small deformations of relatively stiff robot components can negatively impact the precision and the robot functional operation. Such oscillations, which cannot be ignored, are modeled in this study using the finite element (FE) floating frame of reference (FFR) formulation which employs two coupled sets of coordinates; the reference and elastic coordinates. The inverse dynamics, based on the widely used FFR formulation, leads to driving forces associated with the deformation degrees of freedom. Because of the link flexibility, two approaches can be considered in order to determine the actuation forces required to achieve the desired motion trajectories. These two approaches are the partially-constrained inverse dynamics (PCID) and the fully-constrained inverse dynamics (FCID). The FCID approach, which will be considered in future investigations and allows for motion and shape control, can be used to achieve the desired motion trajectories and suppress undesirable oscillations. The new small-deformation PCID approach introduced in this study, on the other hand, allows for achieving the desired motion trajectories, determining systematically the actuation forces and moments associated with the robot joint and elastic degrees of freedom, and avoiding deteriorations in the vibration characteristics as measured by the differences between the inverse- and forward-dynamics solutions. To this end, a procedure for determining the actuation forces associated with the deformation degrees of freedom is proposed and is exemplified using piezoelectric actuators.

Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

We study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

Deformation Law and Control Measures of Gob-Side Entry Filled with Gangue in Deep Gobs: A Case Study

Aiming at the large deformation problem of gob-side entry in solid filling mining, the roof subsidence of gob-side entry retaining (GER) was studied under the influence of gangue filling, by taking a deep filling working face in Shandong Province as the engineering background and using theoretical derivation as well as FLAC3D numerical simulation. Research shows that the stiffness of the gangue filling body in the gob and the stiffness and width of the entry protection coal and rock mass (EPCARM) are positively correlated with the GER roof subsidence, which is much less affected by the EPCARM parameters than by the GER stiffness. The GER failure to meet the application requirements is mainly attributed to the insufficient stiffness of the gangue filling body and excessive advance subsidence, which inhibit the roof stress transfer. The GER replacement by the gob-side entry driving (GED) scheme, which implies replacing the entry protection gangue bag wall with the coal pillar with a width of 5 m, will reduce the roof subsidence to 0.114 m, according to the proposed equation. The results obtained are considered quite instrumental in deformation control of the gob-side entry filled with gangue, as well as substantiation of GED and GER applicability options.

A semi-Lagrangian reproducing kernel particle method with particle-based shock algorithm for explosive welding simulation

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kernel particle method (SL-RKPM) is introduced into the volumetric strain energy to accurately embed the key shock physics in the absence of a mesh or grid, which is shown to also ensure the conservation of linear momentum. For kernel stability, a deformation-dependent anisotropic kernel support update algorithm is proposed, which is shown to capture excessive plastic flow and material separation. A quasi-conforming nodal integration is adopted to avoid the need of updating conforming cells which is tedious in extreme deformations. It is shown that the proposed formulation effectively captures shocks, jet formation, and smooth-to-wavy interface morphology transition with good agreement with experimental results.

$$ T\overline{T} $$-like flows in non-linear electrodynamic theories and S-duality

We investigate the $$ T\overline{T} $$ T T ¯ -like flows for non-linear electrodynamic theories in D(=2n)-dimensional spacetime. Our analysis is restricted to the deformation problem of the classical free action by employing the proposed $$ T\overline{T} $$ T T ¯ operator from a simple integration technique. We show that this flow equation is compatible with $$ T\overline{T} $$ T T ¯ deformation of a scalar field theory in D = 2 and of a non-linear Born-Infeld type theory in D = 4 dimensions. However, our computation discloses that this kind of $$ T\overline{T} $$ T T ¯ flow in higher dimensions is essentially different from deformation that has been derived from the AdS/CFT interpretations. Indeed, the gravity that may be exist as a holographic dual theory of this kind of effective Born-Infeld action is not necessarily an AdS space. As an illustrative investigation in D = 4, we shall also show that our construction for the $$ T\overline{T} $$ T T ¯ operator preserves the original SL(2, ℝ) symmetry of a non-supersymmetric Born-Infeld theory, as well as $$ \mathcal{N} $$ N = 2 supersymmetric model. It is shown that the corresponding SL(2, ℝ) invariant action fixes the relationship between the $$ T\overline{T} $$ T T ¯ operator and quadratic form of the energy-momentum tensor in D = 4.

Experimental study on machining deformation of large scale slender beam with weak stiffness of Ti6Al4V

Large-scale slender beam structures with weak stiffness are widely used in the aviation field. There will be a great deformation problem in machining because the overall stiffness of slender beam parts is lower. Firstly, the cutting mechanism and stability theory of the Ti6Al4V material are analyzed, and then the auxiliary support is carried out according to the machining characteristics of the slender beam structure. The feasibility of the deformation suppression measures for the slender beam is verified by experiments. The experimental analysis shows that on the basis of fulcrum auxiliary support, the filling of paraffin melt material is capable of increasing the damping of the whole system, improving the overall stiffness of the machining system, and inhibiting the chatter effect of machining. This method is effective to greatly improve the accuracy and efficiency during machining of slender beam parts. On the premise of the method of processing support with the combination of fulcrum and paraffin, if the tool wear is effectively controlled, the high precision machining of large-scale slender beams can be realized effectively, and the machining deformation of slender beams can be reduced. Although high speed milling has excellent machining effect on the machining accuracy of titanium alloy materials, severe tool wear is observed during high-speed milling of titanium alloy materials. Therefore, high-speed milling of titanium alloy slender beam is suitable to be carried out in the finishing process, which can effectively control tool wear and improve the machining accuracy of parts. Finally, the process verification of typical weak stiffness slender beam skeleton parts is carried out. Through the theoretical and technical support of the experimental scheme, the machining of large-scale slender beam structure parts with weak stiffness is realized.