Numerical Solution Of Dynamic Problems Research Articles

The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. I. Hyperbolicity

AbstractThe numerical solution of dynamic problems for porous fluid‐saturated solids is often performed with the use of simplified equations known as the u‐p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u‐p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u‐p formulations are focused on the question of how well the u‐p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well‐posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u‐p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well‐posedness for the u‐p approximations. The hyperbolicity conditions derived in this paper for the u‐p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations.

Исследование амплитудно-частотных характеристик вязкоупругого тела типа плотины при установившихся вынужденных гармонических колебаниях

The article discusses the amplitude-frequency response of a viscoelastic body of the dam type, under steady-state forced harmonic vibrations. An important factor is the determination of the number of frequencies and resonance peaks that arise in the process of harmonic effect of the water body on the dam. The use of the finite element method (FEM) for the numerical solution of dynamic problems allows, by expanding the solution in terms of eigenmodes and frequencies, to reduce the original problem to a system of separated linear integro-differential equations with respect to the sought parameters of generalized functions. The process of the influence of viscoelastic properties of the dam material on the resonance curves that arise under harmonic loads of different frequencies at different dimensions of the foot of the dam is investigated. The analysis of the curves of the amplitude-frequency responses of a dam-type viscoelastic body, under steady-state forced harmonic oscillations, showed that the occurrence of resonance peaks depends on the viscoelastic properties of the dam body and the dimensions of the foot of the dam. The main resonance peaks occur at frequencies less than the sixth eigenfrequency, as a result of which a further increase in the number of eigenmodes in the expansion does not introduce any significant changes in the amplitude of the distribution of resonance curves of the amplitude-frequency response of the dam.

Novel frictional algorithm implementing angular increment for two-dimensional frictional systems

This study proposes a new frictional algorithm that implements angular increment. The proposed algorithm is used to solve the numerical solutions of dynamic problems in two-dimensional frictional systems. It can accurately obtain the motion responses of a lumped mass under time-varying external forces, and it can compensate for the shortcomings of the numerical frictional algorithm that implements a time step. Specifically, the proposed algorithm 1) overcomes the difficulties encountered when the angles between resultant tangential forces and slip motion are infinitely close, 2) provides accurate solutions for two-dimensional systems under fierce planar motions, and 3) calculates the responses of the mass within a reasonable period. We compare the computation accuracy, efficiency, and robustness of the proposed frictional algorithm and the previous frictional algorithm [1] through several representative scenarios. We reveal that the proposed algorithm has superior computation accuracy, efficiency, and robustness for two-dimensional frictional problems involving slip/stick transitions and sharp bending.

Continual models of layered and block media with slippage and delamination

Continual models of solid media with a discrete set of slip planes (layered, block media) and with nonlinear viscous-plastic type slip conditions at the contact boundaries of structural elements are constructed. The resulting system of equations contains a small viscosity parameter in the denominator of nonlinear free terms. For a stable numerical solution of a system of differential equations, an explicit-implicit method is proposed with an explicit approximation of the equations of motion and an implicit approximation of the constitutive relations containing a small parameter. Examples of numerical solutions of dynamic problems on the scattering of waves on layered cluster in an elastic medium, on the formation of slip and delamination zones at the boundaries of structural elements are given.

Чисельний розв’язок динамічних задач теорії тришарових циліндричних оболонок еліптичного перерізу з поперечним ребристим наповнювачем

Чисельний розв’язок динамічних задач теорії тришарових циліндричних оболонок еліптичного перерізу з поперечним ребристим наповнювачем

Estimation of wave responses from subvertical macrofracture systems using a grid characteristic method

This work investigates the formation and propagation of scattered waves forming a response of crack structures on the seismologic record. The initial impulse is a flat wave front propagating into the depth of the medium. The paper considers the periodic structure of the scattered wave response from a system (cluster) of subvertical cracks. The procedures of estimation of geometrical parameters of such crack structures are based on numerical experiments. We use a grid-characteristic method to calculate the triangular grids with the formulation of boundary conditions at the interface of the medium and the cracks, as well as on the borders of the region of integration with the characteristic properties of the governing equations of the hyperbolic type. This numerical method makes it possible to build the most correct numerical algorithms on the boundaries of the integration domain and on the surfaces of the media division (interfaces) and to take into account the domain of the solution’s dependence and the physics of the problem (propagation of disturbances on the characteristic directions). This is why this method is perhaps the most suitable one for the numerical solution of dynamic problems with a distinct wave character in geologically substantially inhomogeneous media, in particular, for the considered problem of seismic waves interaction with fissured structures.

A generalized Cosserat point element (CPE) for isotropic nonlinear elastic materials including irregular 3-D brick and thin structures

A generalized form for the strain energy of inhomogeneous deformations is developed for a 3-D brick Cosserat Point Element (CPE) which includes full coupling of bending and torsional modes of deformation. The constitutive coefficients, which depend on the reference geometry of the element, are determined by solving eighteen bending problems and six torsion problems on special elements that are parallelepipeds with two right angles. The resulting constitutive coefficients ensure that the strain energy for inhomogeneous deformations remains a positive definite function of the inhomogeneous strain measures for all reference element shapes. A number of example problems are considered which show that the generalized CPE produces results as accurate as enhanced strain and incompatible elements for thin structures and is free of hourglass instabilities typically predicted by these enhanced elements in regions experiencing combined high compression with bending. Recently, Nadler and Rubin [2003] developed a 3-D eight noded brick element based on the theory of a Cosserat point [Rubin 1995; 2000]. This Cosserat point element (CPE) can be used to formulate the numerical solution of dynamic problems for nonlinear hyperelastic materials. The kinematics of the CPE are characterized by eight element director vectors and the kinetics propose eight balance laws of director momentum to determine the dynamic response of the element. The locations of the nodes in the current deformed configuration are characterized by eight nodal director vectors and the element directors are related to the nodal directors by standard tri-linear shape functions. Moreover, the CPE theory considers the element as a structure and introduces a strain energy function which characterizes the response of the structure. Also, the nodal forces are related to derivatives of the strain energy function through algebraic relations in a similar manner to the relationship of the stress to derivatives of the strain energy function in the full three-dimensional theory of hyperelastic materials. The CPE can easily be implemented into standard finite element programs. Specifically, the nodal positions (nodal director vectors) are used as input variables to determine the nodal forces and tangent stiffness as output variables. Here, the element assembly and solution procedures in the computer program FEAP [Taylor 2005] were used to satisfy the equilibrium equations by updating the present nodal director vectors. In the standard finite element procedures for hyperelastic materials the response of the element is determined by integrals over the element region which assume that the kinematic approximation is valid pointwise. This is in contrast with the CPE which needs no integration over the element region. In

A constrained theory of a Cosserat point for the numerical solution of dynamic problems of non-linear elastic rods with rigid cross-sections

A constrained theory of a Cosserat point has been developed for the numerical solution of non-linear elastic rods. The cross-sections of the rod element are constrained to remain rigid but tangential shear deformations and axial extension are admitted. As opposed to the more general theory with deformable cross-sections, the kinetic coupling equations in the numerical formulation of the constrained theory are expressed in terms of the simple physical quantities of force and mechanical moment applied to the common ends of neighboring elements. Also, in contrast with standard finite element methods, the Cosserat element uses a direct approach to the development of constitutive equations. Specifically the kinetic quantities are determined by algebraic expressions which are obtained by derivatives of a strain energy function. Most importantly, no integration is needed over the element region. A number of example problems have been considered which indicate that the constrained Cosserat element can be used to model large deformation dynamic response of non-linear elastic rods.

On numerical solution of dynamic problems in the theory of reinforced shells

Dynamic problems for cylindrical shells reinforced with discrete ribs are examined. A numerical algorithm based on Richardson extrapolation is developed. Specific problems are solved, and the results are analyzed

Numerical Solution of Dynamic Problems for Reinforced Ellipsoidal Shells under Nonstationary Loads

A dynamic problem for reinforced ellipsoidal shells under nonstationary loads is formulated and solved numerically. The results obtained are analyzed

Parametrized formulations of Hamilton's law for numerical solutions of dynamic problems: Part II. Time finite element approximation

In part I of this paper, we presented a consistent mathematical perspective to the formulations of Hamilton's law and unified the formulations by parametrized form with global approximation. In part II of this paper, we extend the formulations to a proper form to develop high-performance time finite elements for numerical solutions of dynamic problems. The two-field mixed formulations are emphasized and the particular features of using lower order interpolation functions are discussed.

Parametrized formulations of Hamilton's law for numerical solutions of dynamic problems: Part I. Global approximation

Many dynamic problems can be solved numerically by using Hamilton's law. The solution is expressed as a series in the time domain with undetermined coefficients. The unknown coefficients are determined by satisfying the Hamilton's law when the solution is allowed to have certain types of variations. The advantage of the method is that it can directly generate a set of algebraic equations without considering the dynamic equilibrium or the governing differential equations. In this paper, the essential features of the Hamilton's law and its variations are re-examined from the numerical perspectives. A general version of variation is proposed and the parametrized formulations are presented. The parametrized formulations unify conventional formulations and also yield many new ones. Illustrative numerical examples in this paper demonstrate that the conventional formulations may not be optimal although they may be rational.

On numerical solution of dynamic problems in the nonlinear theory of reinforced shells

On numerical solution of dynamic problems in the nonlinear theory of reinforced shells

Boundary Element Methods in Dynamic Analysis

A review of boundary element methods for the numerical solution of dynamic problems of linear elasticity is presented. The integral formulation and the corresponding numerical solution of three- and two-dimensional elastodynamics from the direct boundary element method viewpoint and in both the frequency and time domains are described. The special case of the anti-plane motion governed by the scalar wave equation is also considered. In all the cases both harmonic and transient dynamic disturbances are taken into account. Special features of material behavior such as viscoelasticity, inhomogeneity, anisotropy, and poroelasticity are briefly discussed. Some other nonconventional boundary element methods as well as the hybrid scheme that results from the combination of boundary and finite elements are also reviewed. All these boundary element methodologies are applied to: soil-structure interaction problems that include the dynamic analysis of underground and above-ground structures, foundations, piles, and vibration isolation devices; problems of crack propagation and wave diffraction by cracks; and problems dealing with the dynamics of beams, plates, and shells. Finally, a brief assessment of the progress achieved so far in dynamic analysis is made and areas where further research is needed are identified.

Application of the splitting method to the numerical solution of dynamic problems in elasticity theory

Application of the splitting method to the numerical solution of dynamic problems in elasticity theory