Near conserving energy numerical schemes for two-dimensional coupled seismic wave equations

Two-dimensional coupled seismic waves, satisfying the equations of linear isotropic elasticity, on a rectangular domain with initial conditions and periodic boundary conditions, are considered. A quantity conserved by the solution of the continuous problem is used to check the numerical solution of the problem. Second order spatial derivatives, in the x direction, in the y direction and mixed derivative, are approximated by finite differences on a uniform grid. The ordinary second order in time system obtained is transformed into a first order in time system in the displacement and velocity vectors. For the time integration of this system, second order and fourth order exponential splitting methods, which are geometric integrators, are proposed. These explicit splitting methods are not unconditionally stable and the stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.MTM2015-66837-P del Ministerio de Economía y Competitivida

Energy-conserving 3D elastic wave simulation with finite difference discretization on staggered grids with nonconforming interfaces

In this work, we describe an approach to stably simulate the 3D isotropic elastic wave propagation using finite difference discretization on staggered grids with nonconforming interfaces. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by planar interfaces. This discretization setting is motivated by the observation that wave speeds of earth media tend to increase with depth due to sedimentation and consolidation processes. We demonstrate that the layer-wise finite difference discretization approach has the potential to significantly reduce the simulation cost, compared to its counterpart that uses holistically uniform grids. Such discretizations are enabled by summation-by-parts finite difference operators, which are standard finite difference operators with special adaptations near boundaries or interfaces, and simultaneous approximation terms, which are penalty terms appended to the discretized system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized system is shown to preserve the energy-conserving property of the continuous elastic wave equation, and a fortiori ensure the stability of the simulation. Numerical examples are presented to corroborate these analytical developments

Modeling of frame structures undergoing large deformations and large rotations

Numerical simulation of large-scale problems in structural dynamics, such as structures subject to extreme loads, can provide useful insights into structural behavior while minimizing the need for expensive experimental testing for the same. These types of problems are highly non-linear and usually involve material damage, large deformations and sometimes even collapse of structures. Conventionally, frame structures have been modeled using beam-frame finite elements in almost all structural analysis software currently being used by researchers and the industry. However, there are certain limitations associated with this modeling approach. This research focuses on two issues, in particular, of modeling frame structures undergoing large deformations and rotations when subject to extreme loads such as high intensity earthquakes. One of the issues with using beam-frame models is that the theoretical formulation and numerical implementation of such models are rather complicated and are not well understood by the average engineer using such computer programs. The complications arise primarily due the non-additive nature of three dimensional rotational degrees of freedom, especially under large rotations. Further, ensuring that the time integration schemes used for such models provide stable and accurate solutions is still an active and challenging area of research. To address this issue, a reduced order model that idealizes a frame structure as a network of rotational and extensional springs is developed. This formulation eliminates all the rotational degrees of freedom in the system by expressing the force-displacement and moment-rotation relationships only in terms of nodal coordinates. This not only simplifies the formulation, making it similar in complexity to a network of truss elements, but also avoids the numerous implementational hurdles associated with large three dimensional rotations. Several numerical examples are presented to verify and validate the performance of this approach against conventional beam-frame elements. Existing models that attempt to capture the non-linear behavior of structures undergoing large deformations and damage, which often occurs across multiple scales of space and time, are either limited in the level of fidelity they offer or have an extremely high computational cost associated with them. A computationally advantageous way of approaching such problems is to decompose the structural domain into two regions, one comprising most structural elements where beam-frame elements can be used, and the other consisting of joint and connection regions where more detailed continuum elements can be used as needed. This allows one to model the critical structural components with great fidelity, while still using beam elements for the rest of the model to keep the total computational cost in check. An essential ingredient for this approach is the formulation of a geometrically consistent coupling of beams and continuum elements, especially in the presence of large deformations and large rotations. In addition to spatial coupling of beam and continuum elements, a multi-time-step method is also formulated to allow the beam and continuum elements to be simulated at different time scales. This further adds to the computational efficiency of this approach. Numerical characteristics of such coupled models are verified with a variety of static and dynamic benchmark problems

Dynamic analysis of beam structures considering geometric and constitutive nonlinearity

A fully geometric and constitutive nonlinear model for the description of the dynamic behavior of beam structures is developed. The proposed formulation is based on the geometrically exact formulation for beams due to Simo but, in this article an intermediate curved reference configuration is considered. The resulting deformation map belongs to a nonlinear differential manifold and, therefore, an appropriated version of Newmark’s scheme is used in updating the kinematics variables. Each material point of the cross-section is assumed to be composed of several simple materials with their own constitutive laws. The mixing rule is used to describe the resulting composite. An explicit expression for the objective measure of the strain rate acting on each material point is deduced in this article. Details about its numerical implementation in the time-stepping scheme are also addressed. Viscosity is included at the constitutive level by means of a thermodynamically consistent visco damage model developed in terms of the material description of the First Piola Kirchhoff stress vector. The constitutive part of the tangent tensor is deduced including the effect of rate dependent inelasticity. Finally, several numerical examples, validating the proposed formulation, are given

Modeling three-dimensional acoustic propagation in underwater waveguides using the longitudinally invariant finite element method

textThree-dimensional acoustic propagation in shallow water waveguides is studied using the longitudinally invariant finite element method. This technique is appropriate for environments with lateral variations that occur in only one dimension. In this method, a transform is applied to the three-dimensional Helmholtz equation to remove the range-independent dimension. The finite element method is employed to solve the transformed Helmholtz equation for each out-of-plane wavenumber. Finally, the inverse transform is used to transform the pressure field back to three-dimensional spatial coordinates. Due to the oscillatory nature of the inverse transform, two integration techniques are developed. The first is a Riemann sum combined with a wavenumber sampling method that efficiently captures the essential components of the integrand. The other is a modified adaptive Clenshaw-Curtis quadrature. Three-dimensional transmission loss is computed for a Pekeris waveguide, underwater wedge, and Gaussian canyon. For each waveguide, the two integration schemes are compared in terms of accuracy and efficiency.Mechanical Engineerin

Numerical modeling of 1-D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot's model with frequency-independant coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space-time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.Comment: submitted to the Journal of Computational and Applied Mathematics (2008