Scipedia: 16th World Congress on Computational Mechanics and 4th Pan American Congress on Computational Mechanics

3D-ACA for the time domain boundary element method: Comparison of FMM and H-matrix based approaches

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:08:07 +0200

The time domain Boundary Element Method (BEM) for the homogeneous wave equation with vanishing initial conditions is considered. The generalized convolution quadrature method (gCQ) developed by Lopez-Fernandez and Sauter is used for the temporal discretisation. The spatial discretisation is done classically using low order shape functions. A collocation approach is applied. Essentially, the gCQ requires to establish boundary element matrices of the corresponding elliptic problem in Laplace domain at several complex frequencies. Consequently, an array of system matrices is obtained. This array of system matrices can be interpreted as a threedimensional array of data which should be approximated by a data-sparse representation. The generalised Adaptive Cross Approximation (3D-ACA) can be applied to get a data sparse representation of these three-dimensional data arrays. Adaptively, the rank of the three-dimensional data array is increased until a prescribed accuracy is obtained. On a pure algebraic level it is decided whether a low-rank approximation of the three-dimensional data array is close enough to the original matrix. Within the data slices corresponding to the BEM calculations at each frequency either the standard H-matrices approach with ACA or a fast multipole (FMM) approach can be used. The third dimension of the data array represents the complex frequencies. Hence, the algorithm makes not only a data sparse approximation in the two spatial dimensions but detects adaptively how much frequencies are necessary for which matrix block. Numerical studies show the performance of these methods

Band gap evolution in nonlinear dynamics of metamaterials made structures via gradually-changing mechanical properties

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:08:44 +0200

This paper explores the dynamic behavior of metamaterial-like structures by investigating the evolution of their band gap under the influence of geometrical nonlinearities in the large displacement/rotations field. The study employs a unified framework based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach to develop higher-order onedimensional beam theories that account for geometric nonlinearities. The axis discretization is achieved through a finite element approximation. The equations of motion are solved around nonlinear static equilibrium states, which are determined using a Newton–Raphson algorithm combined with a path-following method of arc-length type. The CUF approach introduces two key innovations that are highly suitable for the evolution of the band gap: 1) Thin-walled structures can be effectively represented using a single one-dimensional beam model, overcoming the common limitations of standard finite elements. This is crucial as three-dimensional solid elements would result in significant computational costs, and twodimensional elements pose limitations for this type of investigation. Finally, employing onedimensional finite elements usually requires a combination of elements, leading to additional mathematical complexities in their connections and lacking geometric precision. 2) CUF enables the use of the full Green-Lagrange strain tensor without the need for assumptions, as is the case with von Karm´ an nonlinearities. ´ The paper specifically compares results obtained with linear and nonlinear stiffness matrices, highlighting the differences. Numerical investigations are conducted on thin-walled structures composed of repeatable cells, assessing mode changes under traction and compression loading. The findings emphasize that the band gap is an inherent property of the equilibrium state, underscoring the necessity of a proper nonlinear analysis for accurately evaluating frequency transitions

A primal hybrid finite element method to solve general compressible, quasi-incompressible and incompressible elasticity using stable H(div)-L2 spaces

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:09:12 +0200

Hybrid methods are usually derived from an extended variational principle, in which the interelement continuity of the functions subspace is removed and weakly enforced by means of a Lagrange multiplier. In this context, a new primal hybrid finite element formulation is presented, which uses H(div) conforming displacement functions and discontinuos L2 approximation for pressure together with shear traction functions to weakly enforce tangential displacement. This combination allows the simulation of compressible, quasi-incompressible and fully incompressible elastic solids, with convergence rates independent of its bulk modulus. The proposed approach benefits from the property that the divergence of the H(div) displacement functions is De Rham compatible with the (dual) pressure functions. The hybridization of the tangential displacements is weakly enforced through a lower order shear stress space. This leads to a saddle-point problem that is stable over the full range of poisson coefficient (large compressibility up to incompressible). Moreover, a boundary stress (normal and shear) can be recovered that satisfies elementwise equilibrium. Hybridizing the tangent stresses and condensing the internal degrees of freedom, a positive-definite matrix with improved spectral properties can be recovered. The stability, consistency and local conservation features are discussed in details. The formulation is tested and verified for different test cases.

Complete variable kinematic cuf-based multilayered shell elements

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:08:57 +0200

The paper presents a methodology for formulating multi-layered composite shell theories with arbitrary kinematic fields. Each displacement variable is examined through an independent expansion function, allowing integration of equivalent single layer and layer-wise approaches within the Carrera Unified Formulation. Finite element method discretizes the structure in the reference plane of the plate using Lagrange-based elements. Governing equations are derived using the principle of virtual displacements. The study considers multilayered structures with different radius-to-thickness ratios and compares results with analytical solutions from the literature. Findings suggest the most appropriate model selection depends strongly on specific problem parameters

A compact sixth order finite difference scheme for the 3D Poisson equation

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:08:33 +0200

Discretization error estimation for flow simulations using general hybrid grids

Jesús Sánchez Pinedo — Fri, 28 Jun 2024 11:08:19 +0200

A primary area of the author’s work with his students is outlined in the present article. It regards the estimation of the discretization error with mixed-element and adaptive meshes. Use of general hybrid meshes for computational flow simulations is of importance due to the complexity of both the geometry and the fields. The meshes can consist of a mix of hexahedra, prisms and tetrahedra with pyramids being transitional elements. The discretization error is a primary component of the numerical error in flow simulations. Primary factors affecting it are the local density of the mesh, as well as its “distortions”, namely the variation in local size and orientation (stretching, skewness), the shape of the individual elements (shear, twist), and the local change in their type (grid interfaces). Two distinct approaches have been followed in order to estimate and control the discretization error. The grid-based (“a priori”) approach assesses mesh quality from the analytic expression of the truncation error. The solution-based (“a posteriori”) approach monitors approximations of the variation of flow quantities (“sensors”). Those are then applied to guide adaptation of the grid to the simulated flow field