The Fixed-Mesh ALE approach for the numerical approximation of flows in moving domains Ramon Codina 1,∗ , Guillaume Houzeaux 2 , Herbert Coppola-Owen 1 and Joan Baiges 1 1 International Center for Numerical Methods in Engineering (CIMNE), Universitat Polit` ecnica de Catalunya, Jordi Girona 1-3, Edifici C1, 08034 Barcelona, Spain. 2 Barcelona Supercomputing Center, Jordi Girona 29, Edifici Nexus II, 08034 Barcelona, Spain. * ramon.codina@upc.edu Contents 1 Introduction 2 2 The Fixed-Mesh ALE method 3 2.1 The classical ALE method and its finite element approximation ........................ 3 2.1.1 Problem statement ............................................ 3 2.1.2 The time-discrete problem ....................................... 4 2.1.3 The fully discrete problem ....................................... 5 2.2 The fixed-mesh ALE approach: algorithmic steps ................................ 7 2.3 Other fixed grid methods ............................................ 8 3 Developing the Fixed-Mesh ALE method 10 3.1 Step 1. Boundary function update ........................................ 10 3.2 Step 2. Mesh velocity .............................................. 10 3.3 Step 3. Solving the flow equations I: Equations on the deformed mesh ..................... 11 3.4 Step 4. Splitting of elements ........................................... 11 3.5 Step 5. Solving the flow equations II: Equations on the background mesh ................... 12 3.6 Comparison with the classical ALE approach .................................. 12 4 Side numerical ingredients 13 4.1 Level set function update ............................................ 13 4.2 Approximate imposition of boundary conditions ................................ 14 4.3 Data transfer between finite element meshes .................................. 16 5 A numerical example 17 6 Two applications 21 6.1 Lost foam casting ................................................ 21 6.2 Free surface flows ................................................ 23 7 Conclusions 25 1