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The purpose of this paper is to explore the possibility of combining additive manufacturing (AM) with topology optimization to generate support structures for addressing the challenging overhang problem. The overhang problem is considered as a constraint, and a novel algorithm based on continuum topology optimization is proposed.

A mathematical model is formulated, and the overhang constraint is embedded implicitly through a Heaviside function projection. The algorithm is based on the Solid Isotropic Material Penalization (SIMP) method, and the optimization problem is solved through sensitivity analysis.

The overhang problem of the support structures is fixed. The optimal topology of the support structures is developed from a mechanical perspective and remains stable as the material volume of support structures changes, which allows engineers to adjust the material volume to save cost and printing time and meanwhile ensure sufficient stiffness of the support structures. Three types of load conditions for practical application are considered. By discussing the uniform distributive load condition, a compromise result is achieved. By discussing the point load condition, the removal work of support structures after printing is alleviated. By discussing the most unfavorable load condition, the worst collapse situation of the printing model during printing process is sufficiently considered. Numerical examples show feasibility and effectiveness of the algorithm.

The proposed algorithm involves time-consuming finite element analysis and iterative solution, which increase the computation burden. Only the overhang constraint and the minimum compliance problem are discussed, while other constraints and objective functions may be of interest.

Compared with most of the existing heuristic or geometry-based support-generating algorithms, the proposed algorithm develops support structures for AM from a mechanical perspective, which is necessary for support structures particularly used in AM for mega-scale construction such as architectures and sculptures to ensure printing success and accuracy of the printed model.

With the rapid development of AM, complicated structures result from topology optimization are available for fabrication. The present paper demonstrates a combination of AM and topology optimization, which is the trend of fabricating manner in the future.

This paper remarks the first of attempts to use continuum topology optimization method to generate support structures for AM. The methodology used in this work is theoretically meaningful and conclusions drawn in this paper can be of important instruction value and practical significance.

This study extends the educational institutions' performance and efficiency literature by examining Brazil's Federal Institute of Education, Science and Technology (FIEST), which consists of educational units throughout the country that span several levels of education.

The authors build and analyze a covariance matrix consisting of both a group of efficiency measures and a group of performance indicators used by Brazil's Ministry of Education (BME). The values in the covariance matrix are maximized through application of the Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS), in which the weights of each variable are optimized in order to capture the direction of the relationship between the two sets of efficiency measures.

Although the authors find that the collective efficiency of the educational units analyzed did not change during the period of study, the analysis reveals that government indicators of performance do not exhibit a strong relationship to the ideal solution efficiency measures used in this study.

This study extends the educational institution efficiency literature by examining Brazil's FIEST, which consists of 40 educational units throughout the country that spans several levels of education, from upper high school vocational courses to higher degrees.