Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

High-fidelity surrogate models for parametric shape design in microfluidics

Nowadays, the main computational bottleneck in computer-assisted industrial design procedures is the necessity of testing multiple parameter settings for the same problem. Material properties, boundary conditions or geometry may have a relevant influence on the solution of those problems. Consequently, the effects of changes in these quantities on the numerical solution need to be accurately estimated. That leads to significantly time-consuming multi-query procedures during decision-making processes. Microfluidics is one of the many fields affected by this issue, especially in the context of the design of robotic devices inspired by natural microswimmers. Reduced-order modelling procedures are commonly employed to reduce the computational burden of such parametric studies with multiple parameters. Moreover, highfidelity simulation techniques play a crucial role in the accurate approximation of the flow features appearing in complex geometries. This thesis proposes a coupled methodology based on the high-order hybridisable discontinuous Galerkin (HDG) method and the proper generalized decomposition (PGD) technique. Geometrically parametrised Stokes equations are solved exploiting the innovative HDG-PGD framework. On the one hand, the parameters describing the geometry of the domain act as extra-coordinates and PGD is employed to construct a separated approximation of the solution. On the other hand, HDG mixed formulation allows separating exactly the terms introduced by the parametric mapping into products of functions depending either on the spatial or on the parametric unknowns. Convergence results validate the methodology and more realistic test cases, inspired by microswimmer devices involving variable geometries, show the potential of the proposed HDG-PGD framework in parametric shape design. The PGD-based surrogate models are also utilised to construct separated response surfaces for the drag force. A comparison between response surfaces obtained through the apriori and the a posteriori PGD is exposed. A critical analysis of the two techniques is presented reporting advantages and drawbacks of both in terms of computational costs and accuracy.Actualmente, el principal obstáculo en los procesos de diseño industrial computarizado es la necesidad de examinar múltiples parámetros para el mismo problema. Las propiedades de los materiales, las condiciones de contorno o la geometría pueden tener una influencia relevante en la solución de esos problemas. Por lo tanto, es necesario estimar con precisión los efectos de las variaciones de esas cantidades en la solución numérica. Esto da origen a procedimientos de consultas múltiples que requieren considerable tiempo durante los procesos de toma de decisión. La microfluídica es uno de los varios campos afectados por esta problemática, especialmente en el contexto del diseño de dispositivos robóticos inspirados en los micronadadores naturales. Generalmente se recurre a procedimientos de reducción de orden de modelo para reducir la complejidad computacional de estos estudios paramétricos basados en múltiples parámetros. Además, los esquemas de alto orden son fundamentales para la aproximación precisa de las particularidades de los flujos que aparecen en las geometrías complejas. Esta tesis propone una metodología acoplada basada en el método de Galerkin discontinuo hibridizable de alto orden (HDG) y la técnica de descomposición propia generalizada (PGD). Las ecuaciones de Stokes geométricamente parametrizadas se resuelven empleando el innovador método HDG-PGD. Por un lado, los parámetros que describen la geometría del dominio actúan como extra-coordinadas y la PGD permite construir una aproximación separada de la solución. Por otra parte, la formulación mixta de HDG admite la separación exacta de los términos introducidos por la descripción paramétrica del dominio en productos de funciones dependientes de las incógnitas espaciales o paramétricas. Los resultados de convergencia validan la metodología y estudios de casos más realistas, inspirados en los dispositivos de micronatación con geometrías variables, muestran el potencial del marco propuesto de HDG-PGD en el diseño de formas parametrizadas. Los modelos reducidos basados en la PGD también permiten construir superficies de respuesta separadas para la fuerza de arrastre. Se realiza una comparación entre las superficies de respuesta obtenidas mediante la PGD a priori y a posteriori. Se exponen una análisis crítica de las dos técnicas reportando las ventajas y desventajas de ambas en términos de costes computacionales y precisión

Adaptivity and Online Basis Construction for Generalized Multiscale Finite Element Methods

Many problems in application involve media with multiple scale, for example, in composite materials, porous media. These problems are usually computationally challenging since fine grid computation is extremely expensive. Therefore, one may need to develop a coarse grid model reduction for this type of problems. In this dissertation, we will consider a multiscale method called generalized multiscale finite element method (GMsFEM). GMsFEM follows the framework of multiscale finite element method. Instead of using one basis function per coarse grid node, GMsFEM uses several basis functions for one coarse grid node. Since the media is highly heterogeneous and may involves high contrast, having more than one basis function per node is important to reduce the error significantly. Due to the varying heterogeneity in the domain, we may require different numbers of basis functions in different regions. Then the question is how to determine the number of basis functions in each region. In this dissertation, we will discuss an adaptive enrichment algorithm for enriching basis functions for the regions with large error. We will consider two different types of basis function for enrichment. One is using the pre-computed offline basis functions. We call this method offline adaptive enrichment. The other method uses online constructed basis functions called online adaptive enrichment. In applications, non-conforming basis functions can give us more flexibility on gridding. The discontinuous Galerkin method also makes the mass matrix block diagonal, which enhances the computation speed in solving time-dependent problem with an explicit scheme. In this dissertation, we will discuss offline and online adaptive methods for the generalized multiscale discontinuous Galerkin method (GMsDGM). We will also discuss using GMsDGM for simulating wave propagation in heterogeneous media

Numerical Simulation of Stresses due to Solid State Transformations : The Simulation of Laser Hardening

The properties of many engineeringmaterialsmay be favourablymodified by application of\ud a suitable heat treatment. Examples are precipitation hardening, tempering and annealing.\ud One of the most important treatments is the transformation hardening of steel. Steel is an\ud alloy of iron and carbon. At room temperature the sollubility of carbon in steel is negligible.\ud The carbon seggregates as cementite (Fe3C). By heating the steel above austenization\ud temperature a crystal structure is obtained in which the carbon does solve. When cooled\ud fast the carbon cannot seggregate. The resulting structure, martensite is very hard and also\ud has good corrosion resistance.\ud Traditionally harding is done by first heating the whole workpiece in an oven and then\ud quenching it in air, oil or water. Other methods such as laser hardening and induction\ud hardening are charaterized by a very localized heat input. The quenching is achieved by\ud thermal conduction to the cold bulk material. A critical factor in these processes is the time\ud required for the carbon to dissolve and homogenize in the austenite.\ud This thesis consists of two parts. In the first part algorithms and methods are developed\ud for simulating phase transformations and the stresses which are generated by inhomogeneous\ud temperature and phase distributions. In particular the integration of the constitutive\ud equations at large time increments is explored. The interactions between temperatures,\ud stresses and phase transformations are cast into constitutive models which are suitable for\ud implementation into a finite element model.\ud The second part is concerned with simulation of steady state laser hardening. Two\ud different methods are elaborated, the Arbtrary Lagrangian Eulerian (ALE) method and a\ud direct steady state method. In the ALE method a transient calculation is prolonged until\ud a steady state is reached. An improvement of the convection algorithm enables to obtain\ud accurate results within acceptable calculation times.\ud In the steady state method the steadiness of the process is directly incorporated into\ud the integration of the constitutive equations. It is a simplified version of a method recently\ud published in the literature. It works well for calculation of temperatures and phase distributions.\ud When applied to the computation of distortions and stresses, the convergence of the\ud method is not yet satisfactory.\ud i