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Generation of spin imbalance in nonmagnetic semiconductors is crucial for the functioning of many spintronic devices. An attractive design of spin injectors into semiconductors is based on a spin pumping from a precessing ferromagnet, typically excited by a microwave magnetic field leading to a high power consumption of the device. Here we describe theoretically a spin injector with greatly reduced energy losses, in which the magnetic dynamics is excited by an elastic wave injected into a ferromagnet-semiconductor heterostructure. To demonstrate the efficient functioning of such an injector, we perform micromagnetoelastic simulations of the coupled elastic and magnetic dynamics in Ni films and Ni/GaAs bilayers. For thick Ni films, it is shown that a monochromatic acoustic wave generates a spin wave with the same frequency and wavelength, which propagates over distances of several micrometers at the excitation frequencies close to the frequency of ferromagnetic resonance. The simulations of Ni/GaAs bilayers with Ni thicknesses comparable to the wavelength of the injected acoustic wave demonstrate the development of a steady-state magnetization precession at the Ni/GaAs interface. The amplitude of such a precession has a maximum at Ni thickness amounting to three quarters of the wavelength of the elastic wave, which is explained by an analytical model. Using simulation data obtained for the magnetization precession at the Ni/GaAs interface, we evaluate the spin current pumped into GaAs and calculate the spin accumulation in it by solving the spin diffusion equation. Then the electrical signals resulting from the spin flow and the inverse spin Hall effect are determined via the numerical solution of the Laplace's equation. It is shown that amplitudes of these ac signals are large enough for experimental measurement, which indicates an efficient acoustically driven spin pumping into GaAs.