### Abstract

We propose a space–time boundary element method for the dynamic simulation of elastic truss systems. The considered truss systems consist of several members, where in each elastic rod the dynamic behaviour is governed by the 1D wave equation. The time domain fundamental solution and boundary integral equations are used to establish the dynamic Dirichlet-to-Neumann map for a single rod. Thus, we are able to reduce the problem to the nodes of the truss system and therefore only a temporal discretization at the truss nodes is necessary. We introduce a stepwise solution strategy with local step size which ensures stability. Furthermore, the discretization within each of these time steps can be refined adaptively to reduce the approximation error efficiently. The optimal convergence of the method is demonstrated in numerical examples. Due to adaptive refinement, this optimal convergence rate is retained even for non-smooth solutions. Finally, the method is applied to study typical truss systems.

Original language | English |
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Pages (from-to) | 37-57 |

Number of pages | 21 |

Journal | Wave Motion |

Volume | 87 |

DOIs | |

Publication status | Published - 2019 |

### Fingerprint

### Keywords

- Adaptive mesh refinement
- Boundary integral equation
- Dirichlet-to-Neumann map
- Stability
- Wave equation

### ASJC Scopus subject areas

- Computational Mathematics
- Physics and Astronomy(all)
- Applied Mathematics
- Modelling and Simulation

### Fields of Expertise

- Information, Communication & Computing

### Cite this

**Wave Propagation in Elastic Trusses: An Approach via Retarded Potentials.** / Pölz, Dominik; Gfrerer, Michael Helmut; Schanz, Martin.

Research output: Contribution to journal › Article › Research › peer-review

}

TY - JOUR

T1 - Wave Propagation in Elastic Trusses: An Approach via Retarded Potentials

AU - Pölz, Dominik

AU - Gfrerer, Michael Helmut

AU - Schanz, Martin

PY - 2019

Y1 - 2019

N2 - We propose a space–time boundary element method for the dynamic simulation of elastic truss systems. The considered truss systems consist of several members, where in each elastic rod the dynamic behaviour is governed by the 1D wave equation. The time domain fundamental solution and boundary integral equations are used to establish the dynamic Dirichlet-to-Neumann map for a single rod. Thus, we are able to reduce the problem to the nodes of the truss system and therefore only a temporal discretization at the truss nodes is necessary. We introduce a stepwise solution strategy with local step size which ensures stability. Furthermore, the discretization within each of these time steps can be refined adaptively to reduce the approximation error efficiently. The optimal convergence of the method is demonstrated in numerical examples. Due to adaptive refinement, this optimal convergence rate is retained even for non-smooth solutions. Finally, the method is applied to study typical truss systems.

AB - We propose a space–time boundary element method for the dynamic simulation of elastic truss systems. The considered truss systems consist of several members, where in each elastic rod the dynamic behaviour is governed by the 1D wave equation. The time domain fundamental solution and boundary integral equations are used to establish the dynamic Dirichlet-to-Neumann map for a single rod. Thus, we are able to reduce the problem to the nodes of the truss system and therefore only a temporal discretization at the truss nodes is necessary. We introduce a stepwise solution strategy with local step size which ensures stability. Furthermore, the discretization within each of these time steps can be refined adaptively to reduce the approximation error efficiently. The optimal convergence of the method is demonstrated in numerical examples. Due to adaptive refinement, this optimal convergence rate is retained even for non-smooth solutions. Finally, the method is applied to study typical truss systems.

KW - Adaptive mesh refinement

KW - Boundary integral equation

KW - Dirichlet-to-Neumann map

KW - Stability

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=85049472907&partnerID=8YFLogxK

U2 - 10.1016/j.wavemoti.2018.06.002

DO - 10.1016/j.wavemoti.2018.06.002

M3 - Article

VL - 87

SP - 37

EP - 57

JO - Wave Motion

JF - Wave Motion

SN - 0165-2125

ER -