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The beam windows of high-energy beam lines are important, and it is sometimes difficult to design because it is necessary to ensure particle propagation with minimum disturbance and fulfill mechanical requirements at the same time. The upstream decay pipe window of the long baseline neutrino facility at Fermilab has an extremely large diameter (1.8 m), with a thickness of only 1.5 mm to separate the helium atmosphere in the decay pipe and the nitrogen atmosphere on the other side. Furthermore, the center of this dish-shaped window is expected to be a 200-mm-diameter beryllium dish welded to the outside aluminum alloy A6061, and this welded combination must withstand extreme conditions of a 2.4-MW, high-energy proton beam without leakage. These severe conditions make the design of this window an unprecedented challenge. This paper describes the static thermal-structural analyses based on which the structure has been optimized, as well as dynamic analyses for understanding the shockwave effects originating in the beam. After optimization, the maximum von Mises stresses in the window decreased significantly in both normal operation and accident cases, making our design very reasonable.