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Symplectic integrators for probability density evolution and structural dynamics applications
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DOI：10.1016/j.probengmech.2026.103953.png)
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Abstract
En 中文
Probability density evolution is widely used in uncertainty analysis and is typically formulated using partial differential equations (PDEs), such as the Liouville equation. Existing numerical methods for solving the Liouville equation do not consider the conserved quantities of the dynamical system, decreasing the uncertainty level of long-term simulations. To solve this problem, we propose a method based on the symplectic integrator to solve the Liouville equation. The Liouville equation is decomposed into two sets of ordinary differential equations and solved using the flow map of ODEs. The properties of probability density evolution of the Hamiltonian system are obtained, and the necessity of adopting a symplectic integrator is discussed. The symplecticity of the structural dynamic equation is evaluated. The flow map is acquired using superposition, and method with the responses caused by unit initial conditions are determined by the symplectic integrator. A numerical procedure is formulated based on the Liouville equation for probability density evolution and statistical moment calculation. Numerical examples are presented, including a one-dimensional spring-mass system and a beam with varying cross-sections. The uncertainty of the dynamic response is computed using the Liouville equation. Different numerical schemes are compared. The result shows that the symplectic integrator provides higher accuracy for calculating the probability density and statistical moments than the non-symplectic integrator. The proposed symplectic integrator method has similar efficiency as the non-symplectic integrator and much higher efficiency than the Monte Carlo method.
Keywords:
Probability density evolution
Symplectic integrator
Liouville equation
Structural dynamics
Uncertainty analysis
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