Higher-order element methods
High-Order Finite Elements have been deeply researched since their inception in the 1970s - see Szabó, Babuška, and Peano [1, 2, 3, 4]. The main advantage of these methods is their fast convergence to accurate solutions. Compared to Low-Order Finite Element Methods which are used in virtually all commercial finite element software packages, a much faster convergence at significantly reduced costs can be attained (see Figure 1).using High-Order Elements.
In this context, Dr Duczek is interested in the performance of different types of shape functions which determine important aspects of each of the high-order methods.
Nowadays, the p-version of the FEM (hierarchic Legendre polynomials) [4], the Spectral Element Method (SEM; Lagrange polynomials) [5, 6], and the Isogeometric Analysis (IGA; B-splines, NURBS) [7] are widely adopted. Each of these methods has individual advantages that make them attractive for certain areas of application.
The p-FEM possess an inherent p-adaptivity feature due to the hierarchic nature of the shape functions, i.e. the polynomial degree of the finite elements can be easily changed from element to element without sacrificing the compatibility between adjacent elements [8].
The distinctive property of the SEM that sets it apart from the other high-order methods is the possibility to straightforwardly diagonalize the mass matrix by numerical integration and still retain the optimal rates of convergence [9]. This property makes the SEM very attractive for dynamical applications including wave propagation analysis, earthquake engineering, impact and crash test simulations, etc.
The exceptional feature of IGA is that it is possible to construct finite elements with higher continuity, i.e. for adjacent elements, not only the displacements themselves are continuous, but also their derivatives. IGA also offers an easy integration of Computer Aided Design software into the simulation pipeline.
Each high-order method must be carefully selected, for its suitability with the intended area of application.
Fictitous domain methods
The fundamental idea of fictitious domain methods (also known as embedded domain methods) is to relieve analysts from the necessity of generating finite element meshes that are geometry conforming [10, 11, 12]. Thus, one of the major bottlenecks of the simulation pipeline can be circumvented. The basic approach is illustrated in Figure 2.
These methods are well suited for materials with complex microstructure and image-based analysis and therefore, applications are useful for hip replacements [13] and metallic foams, to give just two examples [14].
Structural dynamics
Structural dynamics attempts to solve the equations of motion as efficiently as possible using numerical methods. To this end, implicit and explicit time-integration (time-stepping) are typically employed. While, implicit methods are often unconditionally stable, i.e. an arbitrary time step size (time increment) can be used, they involve the solution of a system of equations in each time step.
In contrast, explicit methods are generally conditionally stable, i.e. a critical time increment is provided, but only involve simple matrix-vector multiplications to advance in time. To fully exploit the advantages of explicit methods the mass matrix associated with the spatial discretization has to be formulated in diagonal form [10, 11].
This is, however, not always possible and therefore, alternative methods need to be devised to diagonalize (lump) the mass matrix [9, 12]. Only then can applications such as wave propagation analysis in the context of earthquake engineering, and structural health monitoring be possible.