The paper proposes a density gradient based approach to topology optimization under design-dependent boundary loading. In the density-based topology optimization method, we impose the design dependent loads through spatial gradient of the density. We transform design-dependent boundary loads into a volume form through volume integral of density gradient. In many applications where loadings only need to be exerted on partial boundary, we introduce an auxiliary loading density to keep track of the loading boundary. During the optimization, the loading density is updated by tracking the changes of the physical density in the vicinity of the loading boundary at previous iteration. The proposed approach is easy to implement and computationally efficient. In addition, by adding more auxiliary density fields, the proposed approach is applicable to multiple design-dependent loads. To prevent the intersection of different loading boundaries, a Heaviside projection based integral constraint is developed. Both heat conduction problems under convection loading and elastic problems under hydrostatic pressure loading are presented to illustrate the effectiveness and efficiency of the method.

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