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Heavy ion heating and expansion and laboratory planetary sciences experiments are designed by the high energy density matter generated by heavy ion beams collaboration to study matter under extreme conditions of temperature and pressure at the FAIR facility in Darmstadt, Germany. A special final focusing system has to be installed at the end of the high energy density matter generated by heavy ion beams beam line for strong transverse focusing. To provide a focal spot of the size of a millimeter or less, a large focal angle is needed and, consequently, large-aperture high-gradient quadrupole magnets have to be used in the final focusing system. The Institute for High Energy Physics has considered the basic principles of the development of these magnets. The main parameters of the quadrupole are: the central gradient is 33 T/m; the inner diameter of the coil is 240 mm; the effective length of the magnet is 2 m; and the operating mode is dc. Design features of these quadrupole magnets are detailed: the superconducting wire, 2- and 3-D optimization of the coil and yoke geometry, and mechanical analysis. Preliminary considerations are given on the parameters of the protection and cooling systems of the magnets.