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A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities.

A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation.

Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency.

Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

This study aims to explore the impact of environmental sustainability on government health expenditure in Africa for the period 2000–2021.

The study employs the generalized method of moments (GMM) technique and utilizes a sample of 43 African countries.

The study reveals that carbon dioxide (CO2) emissions and renewable energy consumption have a positive relationship with health expenditure. For the control factors, we demonstrate that economic growth, urbanization and industrialization have a significant negative influence on health expenditure, while the impact of education on health expenditure is positive.

The study proposes several policy recommendations, including prioritizing investment in renewable energy sources and promoting the use of cleaner energy sources such as wind, solar and hydroelectric power. These measures would help improve environmental quality and reduce government health expenditure.

The study contributes to the limited research on the relationship between environmental quality and health expenditure in Africa. Moreover, it goes beyond previous studies that only considered CO2 emissions and explores the impact of adopting cleaner energy sources on health expenditure.