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This paper presents a new method for modeling the mechanics of the aortic valve, and simulates its interaction with blood. As much as possible, the model construction is based on first principles, but such that the model is consistent with experimental observations. We require that tension in the leaflets must support a pressure, then derive a system of partial differential equations governing its mechanical equilibrium. The solution to these differential equations is referred to as the predicted loaded configuration; it includes the loaded leaflet geometry, fiber orientations and tensions needed to support the prescribed load. From this configuration, we derive a reference configuration and constitutive law. In fluid-structure interaction simulations with the immersed boundary method, the model seals reliably under physiological pressures, and opens freely over multiple cardiac cycles. Further, model closure is robust to extreme hypo- and hypertensive pressures. Then, exploiting the unique features of this model construction, we conduct experiments on reference configurations, constitutive laws, and gross morphology. These experiments suggest the following conclusions, which are directly applicable to the design of prosthetic aortic valves. (i) The loaded geometry, tensions and tangent moduli primarily determine model function. (ii) Alterations to the reference configuration have little effect if the predicted loaded configuration is identical. (iii) The leaflets must have sufficiently nonlinear material response to function over a variety of pressures. (iv) Valve performance is highly sensitive to free edge length and leaflet height. For future use, our aortic valve modeling framework offers flexibility in patient-specific models of cardiac flow.