Explicit, implicit, and hybrid methods

Time integration methods can be separated into two groups: explicit and implicit. Methods which do not involve the solution of any algebraic equations are called explicit, while those that require the solution of equations are called implicit. The relative advantages and disadvantages of explicit and implicit methods are summarized. The major trend in the past decade of research was to use hybrization methods to take advantage of the complementary nature of the positive attributes of explicit and implicit integration. These trends are briefly discussed

On the stability of a class of implicit algorithms for nonlinear structural dynamics

Stability in energy for the Newmark beta-family of time integration operators for nonlinear material problems is examined. It is shown that the necessary and sufficient conditions for unconditional stability are equivalent to those predicted by Fourier methods for linear problems

An adaptive continuum/discrete crack approach for meshfree particle methods

A coupled continuum/discrete crack model for strain softening materials is implemented in a meshfree particle code. A coupled damage plasticity constitutive law is applied until a certain strain based threshold value - this is at the maximum tensile stress of the equivalent uniaxial stress strain curve - is reached. At this point a discrete crack is introduced and described as an internal boundary with a traction crack opening relation. Within the frame-work of meshfree particle methods it is possible to model the transition from the continuum to the discrete crack since boundaries and particles can easily be added and removed. The EFG method and an explicit time integration scheme is used. The integrals are evaluated by nodal integration, an integration with stress points and also a full Gauss quadrature. Some results are compared to experimental data and show good agreement. Additional comparisons are made to a pure continuum constitutive law

Efficient linear and nonlinear heat conduction with a quadrilateral element

A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one-point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2, and 1 for the normalized stabilization parameter lead to the 5-point, 9-point finite difference, and fully integrated finite element operators, respectively, for rectangular meshes and have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems

Corotational velocity strain formulations for nonlinear analysis of beams and axisymmetric shells

Finite element formulations for large strain, large displacement problems are formulated using a kinematic description based on the corotational components of the velocity strain. The corotational components are defined in terms of a system that rotates with each element and approximates the rotation of the material. To account for rotations of the material relative to this element system, extra terms are introduced in the velocity strain equations. Although this formulation is incremental, in explicitly integrated transient problems it compares very well with formulations that are not

Probabilistic fracture finite elements

The Probabilistic Fracture Mechanics (PFM) is a promising method for estimating the fatigue life and inspection cycles for mechanical and structural components. The Probability Finite Element Method (PFEM), which is based on second moment analysis, has proved to be a promising, practical approach to handle problems with uncertainties. As the PFEM provides a powerful computational tool to determine first and second moment of random parameters, the second moment reliability method can be easily combined with PFEM to obtain measures of the reliability of the structural system. The method is also being applied to fatigue crack growth. Uncertainties in the material properties of advanced materials such as polycrystalline alloys, ceramics, and composites are commonly observed from experimental tests. This is mainly attributed to intrinsic microcracks, which are randomly distributed as a result of the applied load and the residual stress

Coupling methods for continuum model with molecular model

Coupling methods for continuum models with molecular models are developed. Two methods are studied here: an overlapping domain decomposition method, which has overlapping domain, and an edge-to-edge decomposition method, which has an interface between two models. These two methods enforce the compatibility on the overlapping domain or interface nodes/atoms by the Lagrange multiplier method or the augmented Lagrangian method. 1

A meshfree thin shell method for nonlinear dynamic fracture

A meshfree method for thin shells with finite strains and arbitrary evolving cracks is described. The C1 displacement continuity requirement is met by the approximation, so no special treatments for fulfilling the Kirchhoff condition are necessary. Membrane locking is eliminated by the use of a cubic or quartic polynomial basis. The shell is tested for several elastic and elasto-plastic examples and shows good results. The shell is subsequently extended to modelling cracks. Since no discretization of the director field is needed, the incorporation of discontinuities is easy to implement and straight forwar

Variational approach to probabilistic finite elements

Probabilistic finite element method (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties, and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes