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Element-based topology optimization algorithms capable of generating smooth boundaries have drawn serious attention given the significance of accurate boundary information in engineering applications. The basic framework of a new element-based continuum algorithm is proposed in this paper. This algorithm is based on a smooth-edged material distribution strategy that uses solid/void grid points assigned to each element. Named Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT), the algorithm uses elemental volume fractions which depend on the densities of grid points in the Finite Element Analysis (FEA) model rather than elemental densities. Several numerical examples are studied to demonstrate the application and effectiveness of SEMDOT. In these examples, SEMDOT proved to be capable of obtaining optimized topologies with smooth and clear boundaries showing better or comparable performance compared to other topology optimization methods. Through these examples, first, the advantages of using the Heaviside smooth function are discussed in comparison to the Heaviside step function. Then, the benefits of introducing multiple filtering steps in this algorithm are shown. Finally, comparisons are conducted to exhibit the differences between SEMDOT and some well-established element-based algorithms. The validation of the sensitivity analysis method adopted in SEMDOT is conducted using a typical compliant mechanism design case. In addition, this paper provides the Matlab code of SEMDOT for educational and academic purposes.