This paper is devoted to a new error analysis of nonconforming finite element methods. Compared with the classic error analysis in literature, only weak continuity, the F-E-M-Test for nonconforming finite element spaces, and basic H
 m
 regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed. The analysis is motivated by ideas from a posteriori error estimates and projection average operators. One main ingredient is a novel decomposition for some key average terms on (n − 1)-dimensional faces by introducing a piecewise constant projection, which defines the generalization to more general nonconforming finite elements of the results in literature. The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.

A new a priori error estimate of nonconforming finite element methods

We propose a class of 12 degrees of freedom triangular plate bending elements with quadratic rate of convergence. They may be viewed as the second order Specht triangle, while the Specht triangle is one of the best first order plate bending element. The convergence result is proved under minimal smoothness assumption on the solution. Numerical results for both the smooth solution and nonsmmoth solution confirm the theoretical prediction. Mathematics subject classification: 65N38, 65N30.

/pdf/the-quadratic-specht-triangle-277df7zy13.pdf

The Quadratic Specht Triangle

/pdf/a-family-of-nonconforming-rectangular-elements-for-strain-1pzwitkq86.pdf

A Family of Nonconforming Rectangular Elements for Strain Gradient Elasticity

Climate change, greenhouse gases, and global warming are global issues today. Of course, this global issue cannot be separated from the issue of emissions. Various methods to solve generator scheduling problems by considering emissions or Economic Emission Dispatch (EED) have been published, but not to the extent of calculating the cost to reduce emissions. The main objective of this research is to determine the cost of reducing the emission of electricity generation in Indonesia through solving the EED problem. The method proposed to solve the EED problem is an annealing simulation algorithm and tested using an electrical system of eight generators, four different loads, and five combinations of cost and emission weights. This method is tested with various loads (conditions), and each condition is tested with various combinations of cost weights and emission weights. The obtained results were compared with the results of the calculation of the Cuckoo algorithm, and the whale optimization algorithm. The simulation results show that it costs US$258.81 to reduce 1 ton of emissions. This paper can be used as a material for further consideration for the government and generator providers in making policies related to the operation of power plants by considering emissions. 

Generator Scheduling Optimization Involving Emission to Determine Emission Reduction Costs

This research presents a rigorous and innovative approach, the Homotopy Perturbation Method-Laplace Transform Method (HPM-LTM), implemented in Python, for the efficient solution of linear and nonlinear partial differential equations (PDEs). By combining the Homotopy Perturbation technique (HPM) with the Laplace Transform Method (LTM), our method successfully addresses the significant challenges posed by equations with nonlinear components. Through the utilization of He's polynomials, the HPM-LTM approach effectively handles nonlinear terms, resulting in accurate and reliable solutions. To demonstrate the efficacy of our method, we extensively apply it to five representative PDE scenarios, including heat and wave equations. Our comprehensive results substantiate the remarkable accuracy and reliability of the HPM-LTM approach, highlighting its superiority in comparison to conventional approaches that require restrictive assumptions or discretization, which can introduce round-off errors. Furthermore, our method overcomes the limitations imposed by numerical errors inherent in traditional HPM techniques. The robustness, effectiveness, and adaptability of our proposed approach are further validated by its successful application to a wide range of PDE problems across various fields. This research presents a significant contribution to the development of a powerful computational tool for resolving diverse PDE problems, with particular relevance to the engineering discipline. 

A Comparative Study of Hybrid Analytical and Laplace Transform Approaches for Solving Partial Differential Equations in Python

C0-nonconforming elements for fourth order elliptic problem

To cope with the strong vibration of a supporting structure excited by external loads under operating conditions, and in order to achieve the purpose of vibration reduction by structural optimization through modal modification, a modal modification method was proposed, through structural vibration theory. Subsequently, the search performance of an improved particle swarm optimization method was analyzed before conducting a case study on the structural optimization. Finally, aiming at the problem of strong vibration of gun mount at the time of firing, a finite element model of the gun mount was constructed and the type and natural frequency of the gun vibration in a free state was analyzed. Meanwhile, taking the thickness, height and width of the stiffening structure of the bracket as the design variables, combined with the improved particle swarm algorithm, an optimized mathematical model was developed with the first-order natural frequency of the gun mount as the objective function. The secondary development of Abaqus finite element software by using Python is used as a tool to calculate the optimization model. By virtue of optimation, thickness, width and height of the stiffening structure are 156.4mm, 453.7mm and 238.9mm at the range of [100,600]mm, [100,700]mm, [100,700]mm, respectively, and the base frequency of the gun mount has been increased by 11.3%. The effect is remarkable. doi: 10.5829/ije.2022.35.04a.14

Modal Optimization Design of Supporting Structure Based on the Improved Particle Swarm Algorithm

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C0-nonconforming elements for fourth order elliptic problem

Modal Optimization Design of Supporting Structure Based on the Improved Particle Swarm Algorithm