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Consider the following problem: The initial domain D = [0,1]² deforms with time when the left boundary moves according to (*)

In our example c = 0.00001. Suppose that a monitor function f is given. For instance, we can take f such that it is small near a moving arc given by x² + y² = r(t) where r(t) = 0.5 + 0.1t. Let
d = x² + y² - r(t). Note that the zero set of d is a circular arc centered at the origin. For t < 0.5, the deformation method deforms the initial uniform grid at t = 0 to a grid adapted to this arc by using the monitor function

For t > 0.5, the deformation method continues to deform the adaptive grid to a circular arc with increasing radius specified by r(t) acording to the weight function

Also for t > 1, the left boundary begins to oscillate according to (*). The two main steps are first done to achieve the required grid adaptation specified by the monitor function. Then each grid point in the adapted grid are then moved with velocities relative to the node velocities on the moving boundary with the node velocities on the opposite boundary being equal to zero.