Nonlinear Structural Mechanics Problems Research Articles

A simple extrapolated predictor for overcoming the starting and tracking issues in the arc-length method for nonlinear structural mechanics

This paper presents a simplified implementation of the arc-length method for computing the equilibrium paths of nonlinear structural mechanics problems using the finite element method. In the proposed technique, the predictor is computed by extrapolating the solutions from two previously converged load steps. The extrapolation is a linear combination of the previous solutions; therefore, it is simple and inexpensive. Additionally, the proposed extrapolated predictor also serves as a means for identifying the forward movement along the equilibrium path without the need for any sophisticated techniques commonly employed for explicit tracking. The ability of the proposed technique to successfully compute complex equilibrium paths in static structural mechanics problems is demonstrated using seven numerical examples involving truss, beam-column and shell models. The computed numerical results are in excellent agreement with the reference solutions. The present approach does not require prohibitively small increments for its success.

A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems

We propose a non-intrusive reduced basis (RB) method for parametrized nonlinear partial differential equations (PDEs) that leverages models of different accuracy. From a collection of low-fidelity (LF) snapshots, parameter locations are extracted for the evaluations of high-fidelity (HF) snapshots to recover a reduced basis. Multi-fidelity Gaussian process regression (GPR) is employed to approximate the combination coefficients of the reduced basis. LF data is assimilated either via projection onto an LF basis or via an interpolation approach inspired by bifidelity reconstruction. The correlation between HF and LF data is modeled with hyperparameters whose values are automatically determined in the regression step. The proposed methods not only leverage the assimilated LF data to reduce the cost of the offline phase, but also allow for a fast evaluation during the online stage, independent of the computational cost of neither the low-nor the high-fidelity solution. Numerical studies demonstrate the effectiveness of the proposed approach on manufactured examples and problems in nonlinear structural mechanics. Clear benefits of using lower resolution models rather than reduced physics models are observed in both the basis selection and the regression step. An active learning scheme is used for additional snapshot selection at locations with high error. The speed-up in the online evaluation and the high accuracy of extracted quantities of interest makes the multifidelity RB method a powerful tool for outer-loop applications in engineering, as exemplified in uncertainty quantification.

A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity

The aim of this contribution is the presentation of an adaptive finite element procedure for the solution of geometrically and physically non-linear problems in structural mechanics. Within this context, the attention is mainly directed on the error estimation and hierarchical strategies for mesh refinement and coarsening in the case of finite elasto-plastic deformations. An important but sensitive aspect of adaptation approaches of the space discretization is the calculation of mechanical field variables for the modified mesh. Procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. In order to improve the efficiency as well as the convergence behaviour of the adaptive FE process an approach of data transfer primarily related to nodal values is presented. It is characterized by solving the initial value problem not only at the Gauss points but additionally at the nodes of the elements.

DQFEM Analyses of Static and Dynamic Nonlinear Elastic-Plastic Problems Using a GSR-Based Accelerated Constant Stiffness Equilibrium Iteration Technique

An integrated numerical technique for static and dynamic nonlinear structural problems adopting the equilibrium iteration is proposed. The differential quadrature finite element method (DQFEM), which uses the differential quadrature (DQ) techniques to the finite element discretization, is used to analyze the static and dynamic nonlinear structural mechanics problems. Numerical time integration in conjunction with the use of equilibrium iteration is used to update the response history. The equilibrium iteration can be carried out by the accelerated iteration schemes. The global secant relaxation-based accelerated constant stiffness and diagonal stiffness-based predictor-corrector equilibrium iterations which are efficient and reliable are used for the numerical computations. Sample problems are analyzed. Numerical results demonstrate the algorithm.

Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integrated numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes

Integrated numerical technique for static and dynamic nonlinear structural problems adopting the equilibrium iteration is proposed. The differential quadrature finite element method (DQFEM) which uses the differential quadrature (DQ) techniques to the finite element discretization is used to analyze the static and dynamic nonlinear structural mechanics problems. Numerical time integration in conjunction with the use of equilibrium iteration is used to update the response history. The equilibrium iteration can be carried out by the accelerated iteration schemes. The global secant relaxation-based accelerated constant stiffness and diagonal stiffness-based predictor–corrector equilibrium iterations which are efficient and reliable are used for the numerical computations. Sample problems are analyzed. Numerical results demonstrate the algorithm.

Nonlinear Finite Element Analysis using an Object-Oriented Philosophy – Application to Beam Elements and to the Cosserat Continuum

An Object-Oriented Programming (OOP) frame-work is presented for solving nonlinear structural mechanics problems by means of the Finite Element Method (FEM). Emphasis is placed on engineering applications (geometrically nonlinear beam model, and elastoplastic Cosserat continuum), and OOP is employed as an effective tool, which plays an important role in the FEM treatment of such applications. The implementation is based on computational abstractions of both mathematical and physical concepts associated to structural mechanics problems involving geometrical and material nonlinearities. The overall class organization for nonlinear mechanics modeling is discussed in detail. All the analyses rely on a generic control class where several classical and modern nonlinear solution schemes are available. Examples which explore, demonstrate and validate the main features of the overall computational system are presented and discussed.

An asymptotic numerical algorithm for frictionless contact problems

.Perturbation techniques have been successfully developed to solve problems in non-linear structural mechanics. Based on asymptotic expansions, these techniques lead to analytic representation of the solution branches. In elasticity, when solving contact problems, two non-linearities can occur due to contact constraints and to geometry. The aim of this paper is to propose an asymptotic numerical method for frictionless contact problems. Three examples of 2-D contact problems will be studied to establish the efficiency of our algorithm.

Modeling of containment structures on high performance computers

With recent advances in parallel supercomputers and network-connected workstations, the solution of large scale structural engineering problems, such as containment structures, has now become tractable. High-performance computer architectures, which are usually available at large universities and national laboratories, now can solve large nonlinear problems. At the other end of the spectrum, network connected workstations can be configured to become a distributed-parallel computer. A description of the development of a parallelized finite element computer program for the solution of static nonlinear structural mechanics problems is presented here. Also, a finite element methodology is presented for use in finding the structural capacity of reinforced concrete structures. The method is applicable to both cylindrical and rectilinear geometries. Containment structures for nuclear reactors are the final barrier between released radionuclides and the public. Containment structures are constructed from steel, reinforced concrete, or prestressed concrete. US nuclear reactor containment geometries tend to be cylindrical with elliptical or hemispherical heads. The older Soviet designed reactors do not use a containment building to mitigate the effects of accidents. Instead, they employed a sealed set of rectilinear, interconnected compartments, collectively called the accident localization system (ALS), to reduce the release of radionuclides to the atmosphere during accidents. As an illustrative example, the methodology developed herein is applied to a generic VVER-440/V213 design subjected to internal overpressure.

Multigrid algorithms for solving structural mechanics problems on supercomputers

Multigrid methods are described for solving linear and non-linear structural mechanics problems. The algorithms combine relaxation on a fine mesh with the computation of a coarse mesh correction. Both linear matrix equations and symmetric eigenvalue problems are considered. History-dependent non-linear problems are solved by combining the linear matrix solver with the Newton-Raphson iteration algorithm. Special attention is paid to implementing the multigrid methods on vector and coarse grained parallel computers. The basic approach adopted is to perform all the matrix-vector multiplications on the element level. This results in code that requires a low amount of storage. Several large-scale problems are solved on a Convex C240, an Alliant FX/8, and a CRAY Y-MP to demonstrate the speed and storage requirements of the methods. Speed-ups of up to 3.41 are observed on four processors of the CRAY Y-MP.

Finite element analysis of two-dimensional shear flexible frame structures: Nonlinear analysis

Two curved beam/column elements are developed for the nonlinear pre- and pos tbuckling analysis of free-form frames and arches undergoing arbitrari large deflections (displacements and rotations). For both elements, the formulation of the pertinent matrices is pursued on the basis of modified versions of the variational theorem due to Hellinger and Reissner with the effects of transverse shear deformation and bending-stretching coupling included. Numerical results are presented for a number of well-selected test problems which demonstrate that both elements represent reliable and highly accurate tools for the numerical solution of geometrically nonlinear structural mechanics problems.

A Newton-Lanczos method for solution of non-linear finite element equations

Procedures for the solution of nonlinear algebraic, discrete equations arising from the application of the finite element method to initial-boundary value problems in structural mechanics are described and evaluated. Properties of Newton's method, modified Newton method and quasi-Newton including asymptotic cost estimates and local convergence characteristics, are discussed. The use of Lanczos algorithm to solve the linearized set of equations arising at each iterate of Newton's method, is investigated. The result is a technique whereby the rate of convergence can be varied from superlinear to quadratic by controlling a certain tolerance. The above methods are compared on several quite different nonlinear problems in structural mechanics and the results are encouraging.

Recent advances in reduction methods for nonlinear problems

Status and some recent developments in the application of reduction methods to nonlinear structural mechanics problems are summarized. The aspects of reduction methods discussed herein include: (a) selection of basis vectors in nonlinear static and dynamic problems (b) application of reduction methods in nonlinear static analysis of structures subjected to prescribed edge displacements, and (c) use of reduction methods in conjunction with mixed finite element models. Numerical examples are presented to demonstrate the effectiveness of reduction methods in nonlinear problems. Also, a number of research areas which have high potential for application of reduction methods are identified.

Unilateral contact, elastoplasticity and complementarity with reference to offshore pipeline design

Abstract The purpose of this paper is to provide numerical solutions to the following problems in nonlinear structural mechanics. A slender beam with nonlinear, elastoplastic flexural characteristics is unilaterally supported in the plane by a frictionless, rigid profile. The first problem is to determine the equilibrium configuration under given vertical loads and end traction; the second, seemingly new problem is to determine the changes of the unilaterally supporting profile which reduce the curvatures of the beam below assigned values with the least total cost. Both problems arise in offshore engineering when a pipeline has to be laid on a rough seabottom, but analogous problems are found in other contexts. Approximate solution procedures are developed in view of practical applications in this context by a unified approach resting on suitable discretization (cubic splines, lumped compliance models), complementarity theory and mathematical programming. The former problem (analysis) is dealt with in the range of large displacements; geometric linearity is assumed for the latter optimization problem.

Ultimate Strength Analysis of Beam-Columns by Rayleigh-Ras Method (1st report)

For nonlinear problems in structural mechanics, finite element analysis based on the incremental theory has been well established. But a large computing time and cost make it difficult to use this method effectively in stractural analysis and design.In this paper the authors propose the application of classical Rayleigh-Ritz procedure to structural nonlinear problems including material and geometrical nonlinearities. By using this method, the size of the final stiffness matrix to be handled can be considerably reduced and therefore substantial cut of computing time may be expected, although it depends upon the selection of trial functions, a scheme of numerical energy integrals, and types of problems.As numerical examples ultimate strength analysis of uniaxially or biaxially loaded steel H-columns are shown. The present analysis is based on the incremental theory by the Lagrangian approach, and the most general displacement functions of beam-columns, derived on the assumptions that cross sections are rigid and shearing deformations can be neglected, are used as trial functions. Some initial imperfections, i. e. load eccentricities, initial deflections, and residual stresses are taken into account. The authors believe that the same procedure may be effectively applied to nonlinear dynamic problems of beam-columns.

A comparative study of different numerical solution techniques as applied to a nonlinear structural problem

This paper describes and compares a number of different numerical methods frequently used in solution of nonlinear problems in structural mechanics. A simple mechanical system is chosen. By varying the geometry and the stiffness of the members, different types of nonlinear structural behavior may be simulated. A comparative study of various solution techniques applied to the sample problem is given. The solution paths followed by these methods are illustrated by use of map plots of the strain energy and the total energy of the system. Also, a new method for automatic computation of steps for incremental techniques is suggested.

Non-linear methods of structural analysis

The foundations for the solution of physically nonlinear problems in structural mechanics are reviewed. Both aspects, the mathematical characterisation of the nonlinear behaviour and the numerical solution within the frame of the finite element method are discussed. A number of examples are presented which are relevant to problems of reactor technology. In particular problems of nonlinear elasticity, viscoelasticity, creep and elasto-plasticity are dealt with.