121 research outputs found
High-order implicit time integration scheme based on Pad\'e expansions
A single-step high-order implicit time integration scheme for the solution of
transient and wave propagation problems is presented. It is constructed from
the Pad\'e expansions of the matrix exponential solution of a system of
first-order ordinary differential equations formulated in the state-space. A
computationally efficient scheme is developed exploiting the techniques of
polynomial factorization and partial fractions of rational functions, and by
decoupling the solution for the displacement and velocity vectors. An important
feature of the novel algorithm is that no direct inversion of the mass matrix
is required. From the diagonal Pad\'e expansion of order a time-stepping
scheme of order is developed. Here, each elevation of the accuracy by two
orders results in an additional system of real or complex sparse equations to
be solved. These systems are comparable in complexity to the standard Newmark
method, i.e., the effective system matrix is a linear combination of the static
stiffness, damping, and mass matrices. It is shown that the second-order scheme
is equivalent to Newmark's constant average acceleration method, often also
referred to as trapezoidal rule. The proposed time integrator has been
implemented in MATLAB using the built-in direct linear equation solvers. In
this article, numerical examples featuring nearly one million degrees of
freedom are presented. High-accuracy and efficiency in comparison with common
second-order time integration schemes are observed. The MATLAB-implementation
is available from the authors upon request or from the GitHub repository (to be
added).Comment: 43 pages, 19 figure
Consistent Infinitesimal Finite-Element Cell Method: In-Plane Motion
To calculate the unit-impulse response matrix of an unbounded medium for use in a time-domain analysis of medium-structure interaction, the consistent infinitesimal finite-element cell method is developed. Its derivation is based on the finite-element formulation and on similarity. The limit of the cell width is performed analytically yielding a rigorous representation in the radial direction. The discretization is only performed on the structure-medium interface. Explicit expressions of the coefficient matrices for the in-plane motion of anisotropic material are specified. In contrast to the boundary-element formulation, no fundamental solution is necessary and equilibrium and compatibility on the layer interfaces extending from the structure-medium interface to infinity, if present, are incorporated automatically. Excellent accuracy is achieved for an inhomogeneous semi-infinite wedge and a rectangular foundation embedded in an inhomogeneous half-plane
High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Pad\'e expansions
A single-step high-order implicit time integration scheme with controllable
numerical dissipation at high frequencies is presented for the transient
analysis of structural dynamic problems. The amount of numerical dissipation is
controlled by a user-specified value of the spectral radius in
the high frequency limit. Using this user-specified parameter as a weight
factor, a Pad\'e expansion of the matrix exponential solution of the equation
of motion is constructed by mixing the diagonal and sub-diagonal expansions. An
efficient timestepping scheme is designed where systems of equations, similar
in complexity to the standard Newmark method, are solved recursively. It is
shown that the proposed high-order scheme achieves high-frequency dissipation,
while minimizing low-frequency dissipation and period errors. The effectiveness
of the provided dissipation control and the efficiency of the scheme are
demonstrated by numerical examples. A simple guideline for the choice of the
controlling parameter and time step size is provided. The source codes written
in MATLAB and FORTRAN are available for download at:
https://github.com/ChongminSong/HighOrderTimeIntegration.Comment: 37 pages, 36 figures, 89 equation
A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials
This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials. © 2017, Springer-Verlag GmbH Germany
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