The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

The term “p-version of the finite element method” (FEM) first appeared in a publication by Babuška et al. (1981) where the theoretical foundations of a discretization strategy, in which the finite element mesh is fixed and the polynomial degree p of the elements is progressively increased, were established. That investigation was motivated by research published earlier by Szabó and Mehta (1978) where it was demonstrated on the basis of a two-dimensional problem of linear elastic fracture mechanics that increasing the

The p-Version of the Finite Element and Finite Cell Methods

In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’.

/pdf/a-selection-of-benchmark-problems-in-solid-mechanics-and-4o1oga734z.pdf

A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics

Book Review: Leszek Demkowicz, Computing with hp-adaptive finite elements, Volume 1, One and two dimensional elliptic and Maxwell problems

http://repositorium.sdum.uminho.pt/bitstream/1822/53391/1/art22.pdf

A new staggered algorithm for thermomechanical coupled problems

High-order finite elements are commonly compared with linear element formulations showing that, in terms of the relation between computational time and achievable accuracy, linear element formulations are inferior to high-order elements. On the other hand, the finite element community follows the h-version approach due to its simplicity in implementation. This article compares high-order finite elements based on hierarchic shape functions with low-order mixed element formulations using finite strain hyperelasticity. These comparisons are conducted from the point of view of both accuracy and efficiency as well as highly deformed structures. It also investigates improvements to minimize the overall computational effort such as parallelizing the element assemblage procedure, choosing a starting vector estimator for Newton's method, and investigating the Newton-Chord method. The advantages and disadvantages of both finite element approaches are also discussed.

High‐order finite elements compared to low‐order mixed element formulations

This article addresses a thermo-mechanically coupled problem of thermo-viscoelasticity at finite strains using a monolithic approach. The underlying equations are based on the non-linear transient heat equation, the local equilibrium conditions and the evolution equations of the internal variables. The latter describe the hardening behavior of the material. If the method of vertical lines is applied, its first step-namely the spatial discretization-yields a system of differential-algebraic equations (DAE-system). Here, we employ the p-version of the finite element method based on integrated Legendre polynomials. This can lead to very precise solutions in the spatial domain. In order to be accurate in the time-domain as well, stiffly accurate, diagonally-implicit Runge-Kutta methods are applied to solve the DAE-system yielding a coupled system of non-linear algebraic equations. In this article, the system is solved monolithically by employing the Multilevel-Newton algorithm. Accordingly, a high-order result is obtained in the space and the time domain. The numerical concept is applied to a constitutive model of finite strain thermo-viscoelasticity. Several examples are applied to demonstrate the efficiency and applicability of the numerical scheme. It is especially the transient problems that call for time-adaptive schemes which are naturally embedded in the concept.

A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

On the one hand, high-order quasi-static finite elements have been developed on the basis of isogeometric analysis or hierarchical shape functions based on integrated Legendre-polynomials, which display favorable behavior in comparison to linear element formulations. If we consider constitutive models of evolutionary-type such as elastoplasticity, viscoplasticity or viscoelasticity, only first order methods are employed to integrate the ordinary-differential equations at the Gauss-point level, leading to errors caused by the time-integration. On the other hand, high-order time integration methods have been devised to treat the resulting system of differential-algebraic equations (DAE-system) following the spatial discretization. This, however, is only done using spatial discretizations with linear or quadratic h-elements based on Lagrange polynomials or related mixed element formulations. In this article both approaches are combined using a p-version finite element approach on the basis of hierarchical shape functions and the resulting DAE-system is solved by means of high-order one-step methods. The first step in this treatise is to apply stiffly accurate, diagonally-implicit Runge-Kutta methods combined with the Multilevel-Newton algorithm. Since the computational outlay for computing the tangential stiffness matrix is very high in the case of p-version finite elements, Rosenbrock-type methods are applied as well, which lead to a completely iteration-free technique. We then compare the two approaches. The next step is to investigate three-dimensional examples showing the properties on the basis of a constitutive model in finite strain viscoelasticity.

High-order quasi-static finite element computations in space and time with application to finite strain viscoelasticity

In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the directional deformation is proposed. This leads to a clear division of the deformation and stress states in the direction of anisotropy and a remaining part. The decomposition is explained in detail and a constitutive model of hyper-elasticity is proposed for the case of transversal isotropy, where the behavior of the model is investigated with the aid of simple analytical examples. The model is also investigated for inhomogeneous deformation states using high-order finite elements based on hierarchical shape functions showing the sensitivity of the accuracy of the results in the case of anisotropic media.

ERRATUM: Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p-version finite elements

Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p-version finite elements: Transversal isotropy based on multiplicative decompositions

Torben Netz

Papers

High‐order finite elements compared to low‐order mixed element formulations

A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

High-order quasi-static finite element computations in space and time with application to finite strain viscoelasticity

ERRATUM: Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p-version finite elements

Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p-version finite elements: Transversal isotropy based on multiplicative decompositions