Artificial Stiffness Research Articles

Implementation of the nonlinear imperfect transmission conditions between dissimilar materials into commercial FEM software MSC.Marc using Fortran user subroutine: Plane strain case

Thin soft elasto-plastic interphase adhesively bonded materials are replaced by nonlinear imperfect transmission conditions in commercial FEM software MSC.Marc using Fortran user subroutine. The idea of replacement is based upon iteratively finding artificial stiffness of springs which maps elasto-plastic behaviour of interphase. Plane strain case is considered.

Highly efficient solid and solid-shell finite elements with mixed strain–displacement assumptions specifically set up for explicit dynamic simulations using symbolic programming

The method of incompatible modes is applied to volumetric elements and so-called solid-shell elements with linear interpolation as an alternative to the well known method of enhanced assumed strains in order to reduce artificial stiffness effects. The incompatible modes are directly adjoint to known EAS elements. In the context of explicit time integration, the incompatible degrees of freedom are treated as regular unknowns with associated modal masses and incorporated into the global equations instead of being condensed-out locally decreasing the computational effort. The issue of artificial kinetic energy is briefly discussed. Further the implementation concept applying symbolic programming is described.

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshes do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work [8,19,37,38,44] and recent Lagrangian SGH work [33–35,39,45]. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes [18] that remove a volume error in the PCH discretization. A 2-stage Runge–Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.

Improving the numerical efficiency for 3D enhanced assumed strain finite elements in explicit time integration

AbstractExplicit time integration has proven to be highly suitable for the simulation of very fast processes and highly nonlinear applications. As is well known the time integration algorithm is only stable if the Courant‐Friedrichs‐Lewy criterion is fulfilled leading usually to very small time step sizes. As a result the efficiency of the time integration algorithm is dictated by the computational time of element operations. In structural mechanics low order finite elements are attractive due to their efficiency but introduce artificial stiffness effects. To resolve the so‐called locking effects a transient description – suitable for the solution in the context of explicit time integration – of the method of enhanced assumed strains (EAS) is formulated utilizing a generalization of the method of incompatible modes (IM). The element routines are computationally realised applying symbolic programming. The functionality and achieved efficiency are demonstrated on a numeric example. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The QT Interval Could be a Marker of Subclinical Atherosclerosis in Patients with Type 2 Diabetes

Several studies have shown that prolongation of QT interval predicts the risks of all-cause and cardiovascular mortality. Moreover, a close association was reported between prolongations of QT interval and pulse wave velocity (PWV) in general population without preexisting cardiovascular disease. The aim of this study was to evaluate the relationship between QT interval and PWV in patients with type 2 diabetes without preexisting cardiovascular disease. In a cross-sectional study of 251 Japanese patients with type 2 diabetes, we examined the correlation of the heart rate-corrected QT interval duration (QTc) with PWV or other various parameters, including age, duration of diabetes, body mass index, systolic blood pressure, hemoglobin A1c, serum cholesterol, creatinine, uric acid, potassium, severity of diabetic nephropathy or retinopathy and current treatment of diabetes. The QTc correlated positively with PWV (r=0.268, p<0.0001). Multiple regression analysis demonstrated that the QTc was independently correlated with PWV (β=0.155, p=0.0119), after adjustment for known risk factors. According to the receiver operator characteristic analyses (ROC), the area under the ROC curve (AUC) of QTc and the Framingham 10-year coronary heart disease risk score (FRS) for arterial stiffness were 0.69 and 0.64, respectively. The AUC of QTc for arterial stiffness was similar to that of FRS for artificial stiffness (p=0.6532). In conclusion, the QT interval could be a marker of subclinical atherosclerosis in patients with type 2 diabetes.

A modified Galerkin FEM for 1D Helmholtz equations

A method is presented that aims to eliminate the numerical errors inherent in the standard Galerkin finite element method (GFEM) for solving homogeneous Helmholtz equations. An error analysis of the standard GFEM with linear elements is first performed by using the concept of truncation error in finite difference methods, and then the truncation error expression is obtained. A linear GFEM with an artificial stiffness is proposed to solve the Helmholtz equation after investigating the effect of the error on numerical solution. The proposed method is essentially as straightforward as the standard GFEM and thus requires almost no additional computational effort. Numerical results show that present pollution error decreases by 90% compared with that of the standard GFEM.

Hourglass Control Methods in Numerical Simulation Code RingForm

Hourglass is obviously aroused in cold ring rolling simulation with reduced integration element because of ring vibrating violently and small contact area of ring and rollers. In the developed numerical simulation code named RingForm, two methods are used to control the hourglass. The first method is combination of artificial damping and artificial stiffness. The second is one-point quadrature element with hourglass control, which need not user-defined control parameters. Hourglass energy can be reduced to 3 percent of the internal energy. Using the latter, good elements are acquired, while computational efficiency is obviously lower than the former.

Finite element simulation of acoustic cavitation in the reservoir and effects on dynamic response of concrete dams

A 3D finite element formulation for the dynamic analysis of concrete dams is presented. A mixed Eulerian–Lagrangian formulation is used to simulate fluid–structure system. During severe ground motion, the impounding fluid in the reservoir may separate from the dam and causes forming of micro bubbles. As a result, the compressibility of water is reduced. This nonlinear phenomenon of the reservoir is termed as cavitation. When the direction of ground motion is changed, the micro bubble's region of fluid collapses, and an impact will occur. In order to eliminate the spurious oscillations, which are caused by the impact, a small amount of artificial stiffness proportional damping is added in the fluid domain. To capture cavitation effects a bilinear equation of state is employed and incorporated with finite element formulation of fluid domain. An iterative partitioned method is used to simultaneous time integration of equations of motion of fluid and structure domains. The developed method is validated by testing it against problem for which, there is existing solution. Also the effects of cavitation on dynamic response of Koyna gravity dam and Morrow Point arch dam subjected to the first 6 second of the May 1940 El-Centro, California earthquake, is considered. In order that truly consider the effects of cavitation phenomenon, maximum acceleration was scaled to give an amplitude of 1 g. Obtained results show that impact force caused by cavitation has a small effect on the dynamic response of dam–reservoir systems.

Two Hourglass Control Methods for Dynamic Analysis in Cold Ring Rolling Process

The two methods of artificial damping and artificial stiffness are developed in dynamic explicit FEA code for ring rolling. Advantages and disadvantages of artificial damping method and artificial stiffness method are analyzed. A rectangular section cold ring rolling is simulated. The hourglass energy and the ratio of the hourglass energy to the internal energy are compared between the two hourglass control methods. The hourglass control coefficient criterion of the two methods in the cold ring rolling simulation is proposed.

Free vibration of shallow shells using the Rayleigh—Ritz method and penalty parameters

In the present work, the Rayleigh—Ritz method (RRM) and the penalty function method (PFM) are used to solve vibration problems of shallow shells of rectangular planform with spherical, cylindrical, and hyperbolic paraboloidal geometries with classical boundary conditions. Problems with more complicated constraints can also be solved using the method presented in this work, which consists in developing the stiffness and mass matrices of a completely free shallow shell using the RRM and then all constraints of the shell are defined by the PFM through the use of either artificial stiffness or artificial inertia with either positive or negative values.

Sliding contact conditions using the master–slave approach with application on geometrically non-linear beams

Frictionless sliding conditions between two bodies are usually defined using either the method of Lagrangian multipliers or by prescribing an artificial (penalty) stiffness which resists the penetration at the contact point. Both of these methods impose the condition that the contact force should be normal to the contact surface, with the Lagrangian multiplier or the penalty parameter serving as a measure of this force. In this work, an alternative approach is undertaken: the frictionless sliding condition is defined through a relationship between nodal parameters of the virtual displacements of a discretised principle of virtual work. This method, which does not involve additional force parameters or degrees of freedom, is known as the master–slave or the minimum-set method and is particularly convenient for displacement-based finite-element implementation. The method is analysed in detail in context of bilateral sliding constraints characteristic of prismatic and cylindrical joints in flexible beam assemblies undergoing large overall motion. Two numerical examples are presented and assessed against the results in the literature.

Periodic response to fluid loading of single-degree of freedom systems with stiffness nonlinearity

The paper considers single-degree of freedom systems with a nonlinear restoring force and subjected to periodic fluid loading due to waves and currents. The desired number of displacement harmonics are found in the frequency domain using two alternative iterative schemes both of which are based on the Newton–Raphson method and involve an explicit description of both the nonlinear restoring force and the relative velocity-squared drag loading in the time domain. The first option, called the multi-diagonal Jacobian method (MDJM), uses essentially the full Jacobian, whereas the second option, the single-diagonal Jacobian method (SDJM), uses only the main diagonal of the Jacobian. It is shown that the SDJM involves adding an artificial stiffness and an artificial damping to the system. The methods have been implemented for the particular case of cubic nonlinearity, which corresponds to the Duffing equation with linear viscous damping, and a nonlinear fluid loading which may have a non-zero mean. The results have been validated primarily by noting that the residual load at convergence is negligibly small and, for a selected number of cases, also by comparison with the results of the Runge–Kutta–Nystrom time integration. For the system parameters considered here, both the schemes showed good convergence properties. However, as the SDJM needs much less computation per iteration than the MDJM, it works out to be the method of choice.

An algorithm to incorporate friction in finite-element analysis (FEM) of metal working

In the finite-element analysis of metal working, the majority of the algorithms available require prior knowledge of the direction of metal flow for imposing the frictional force as a boundary condition. Even then, in some cases, this may result in unbalanced forces, leading to numerically unsound conditions. In the present study, the problem is solved by introducing an artificial stiffness (frictional force/displacement) to the node of interest. Results of the simulation of a typical ring compression are compared with experimental results, the agreement being found to be very good.

An algorithm to incorporate friction in finite-element analysis (FEM) of metal working

In the finite-element analysis of metal working, the majority of the algorithms available require prior knowledge of the direction of metal flow for imposing the frictional force as a boundary condition. Even then, in some cases, this may result in unbalanced forces, leading to numerically unsound conditions. In the present study, the problem is solved by introducing an artificial stiffness (frictional force/displacement) to the node of interest. Results of the simulation of a typical ring compression are compared with experimental results, the agreement being found to be very good.

On artificial strain of thin curved elements

AbstractThis paper presents an attempt to clarify further the nature of the overstiffness of thin curved structural elements. It follows the approach presented by Ashwell and co‐workers1,2 who based their study on shape functions of arch elements. Such elements display the same stiff behaviour as complex shell elements but are simpler to study and understand. An examination of several discretizations for a two‐dimensional curved beam‐column element for linear elastic analysis of arches is performed. This examination and the numerical results obtained from these approximations provide a new interpretation of the artificial strains based on the mapping variables of shape functions. Two new methods to reduce the artificial stiffness of the element are proposed.

Stability, accuracy and efficiency of some underintegrated methods in finite element computations

In an attempt to increase computational efficiency in the numerical solution of highly nonlinear problems in solid and fluid mechanics, underintegrated finite element methods have been employed by many analysts. Underintegration refers to the use of a rule of an order lower than that required to integrate polynomial integrands exactly. The main drawback of this technique is related to the production of rank-deficient stiffness matrices, or equivalently an expanded kernel of the governing linear momentum operators. Such a development can introduce numerical instabilities. In order to overcome this difficulty, artificial stiffness or viscosity methods, or other stabilization methods have been proposed. One approach involves the elimination of spurious modes in a postprocessing operation. The present study is concerned with this a posteriori elimination method, taking into account the results which can be expected from it, and some of its possible extensions.