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Previous works of Junker and Hackl (2016) have presented a variational growth approach to topology optimization in which the problem of checkerboarding was suppressed by means of a discontinuous regularization scheme. This approach did not require additional filter techniques and also optimization algorithms were not needed any more. However, growth approaches to topology optimization demand some limitations in order to avoid a global and simultaneous generation of mass. The limitation has been achieved by a rather simple approach with restricted possibilities for controlling. In this contribution, we eliminate this drawback by introducing a Lagrange multiplier to control the total mass within the model space for each iteration step. This enables us to achieve directly controlled growth behavior and even find optimized structures for prescribed structure volumes. Furthermore, a modified growth approach, which we refer to as the Lagrange shift approach, results a numerically stable model that is easy to handle. After the derivation of the approach, we present numerical solutions for different boundary problems that demonstrate the potential of our model. © 2016 Elsevier B.V.