Methods, systems, and apparatus, including medium-encoded computer program products, for computer aided design of physical structures include: obtaining a density-based representation of a modeled object and data specifying a starting element for each of multiple different subsets of elements; processing starting elements having milling depths associated with layers below a top most layer, the processing including, for a current starting element for a current layer, identifying other starting elements that have milling depths associated with a layer above the current layer and are closer to the current starting element than an amount at least equal to a radius of a smallest available milling tool, calculating a maximum angular difference, and moving the milling depth for the element subset of the current starting element to a layer above the current layer, responsive to the maximum angular difference being greater than a threshold, to remove a non-manufacturable corner.

Methods, systems, and apparatus, including medium-encoded computer program products, for computer aided design of physical structures include: obtaining a density-based representation of a modeled object and data specifying a starting element for each of multiple different subsets of elements; processing starting elements having milling depths associated with layers below a top most layer, the processing including, for a current starting element for a current layer, identifying other starting elements that have milling depths associated with a layer above the current layer and are closer to the current starting element than an amount at least equal to a radius of a smallest available milling tool, calculating a maximum angular difference, and moving the milling depth for the element subset of the current starting element to a layer above the current layer, responsive to the maximum angular difference being greater than a threshold, to remove a non-manufacturable corner.

Methods, systems, and apparatus, including medium-encoded computer program products, for computer aided design of physical structures include: obtaining a density-based representation of a modeled object and data specifying a starting element for each of multiple different subsets of elements; processing starting elements having milling depths associated with layers below a top most layer, the processing including, for a current starting element for a current layer, identifying other starting elements that have milling depths associated with a layer above the current layer and are closer to the current starting element than an amount at least equal to a radius of a smallest available milling tool, calculating a maximum angular difference, and moving the milling depth for the element subset of the current starting element to a layer above the current layer, responsive to the maximum angular difference being greater than a threshold, to remove a non-manufacturable corner.

A method determines a path of a tool machining a pocket shape in a workpiece. The pocket shape is defined by a closed curve in an x-y plane. The method determines positions and velocities of a motion of an oscillator having an energy defined by a function that is positive within a boundary of the pocket shape and equals zero on the boundary of the pocket shape. The method determines coordinates of the path of the tool based on corresponding pairs of position and velocity values of the motion, wherein a position value of each pair represents an x coordinate of the path of the tool in the x-y plane, and a velocity value of each pair represents a y coordinate of the path of the tool in the x-y plane.