Part of the design is relatively conventional -- a crank on a crankshaft is connected by a piston in cylinder.
However, that piston isn't directly doing work on air, it's doing work on a hydraulic fluid... the piston alternatly pushes that fluid out of the cylinder, and is pushed back by the fluid.
Connected
to the cylinder is a spherical metal chamber. The hydraulic fluid goes into and out of the "bottom" of this chamber. At the top of the chamber is a component exactly identical to a four stroke engine's head. Sealed to this engine head is a gas filled teflon balloon.
During the intake stroke, atmospheric air pressure pushes on the inside of the balloon, which pushes on the hydraulic fluid, which flows back from the sphere chamber to the cylinder, following the piston, which is moving downward.
During the compression stroke, the piston applies pressure to the hydraulic fluid, forcing the fluid into the sphere, where it applies pressure to the balloon, compressing the air inside.
During the power stroke, the hot compressed gasses push very hard on the inside of the balloon, doing work as they push the hydraulic fluid moves from the sphere to cylinder, pushing hard on the piston.
The exhaust stroke is much like the compression stroke, except that since the exhaust valve is open, the hydraulic pressure is much lower, so the piston needs to do less work.
At all times, the pressure inside the balloon and outside the ballon are equal, so the material of the balloon doesn't need to be mechanically strong. It only needs to be resistant to the high heat it will experience, and not be damaged by repeated expansion and contraction.
Keeping the hydraulic fluid cool is an issue, but if we replace a small amount of hot fluid with cool fluid during each cycle, we can avoid overheating. Since the hydraulic fluid is probably the hottest part of the engine, we can probably omit a conventional water cooling system.