Relationships for the approximation of transient direct and inverse problems with asymptotic kernels

Abstract

Many transient problems in mechanics can be resolved through the use of convolution relationships provided the unit impulse or step kernels are known. Unfortunately, solution of the resulting equations for either the direct or inverse problem often requires the use of specialized numerical methods. To help overcome this potential shortcoming, approximate rules for direct and inverse Laplace transformation were used to modify the step-function based convolution relationships to an algebraically solvable and relatively simple form. The resulting relationships can be applied as a first-order approximation to problems in viscoelasticity and heat flow provided the kernel is of an asymptotic exponential form, materials properties do not vary with temperature or strain, and the underlying excitations are not overly oscillatory in nature. Under these provisions, reasonable agreement was seen between the derived relationships and direct solutions for test case studies involving thermal diffusion of a planar slab and the Voigt/Kelvin viscoelastic model of a spring and dashpot in parallel. Within the accuracy confines of a non-adaptive inverse analysis utilizing a single response history, the method was also shown to be capable of producing reasonable estimates of the underlying excitations for both test cases.

Keywords:

• Transient
• Direct
• Inverse
• Thermal diffusion
• Viscoelasticity
• Laplace transform

^*Tel.: [phone], Fax: [phone], [email protected]

^*Tel.: [phone], Fax: [phone], [email protected]

Notes

^*Tel.: [phone], Fax: [phone], [email protected]