In this work, a topology optimization procedure based on the level-set method is applied to the solution of inverse problems for acoustic wave propagation in the time-domain. In this class of inverse problems the presence of obstacles in a background medium must be identified. Obstacles and background are defined by means of a level-set function that evolves by following the solution of a reaction–diffusion equation. Within this approach, no initial guess for the topology nor level-set reinitialization procedures are necessary, contrary to what is commonly observed when the Hamilton–Jacobi equation is used. The objective function is defined as the domain and time integration of the squared difference between experimental and simulation pressure signals. The finite element method is used for the spatial and level-set function discretizations and a time-marching procedure (Newmark scheme) is used to solve the wave propagation problem, as well as the adjoint problem for the sensitivity analysis. Both procedures provide the information needed to define the velocity field for the level set evolution. Results show that the proposed technique is capable to find the location and shape of obstacles within a background medium. Systematic tests show that, as expected, the distribution of sources and receivers shows to have a major influence on the final solution. Results also reproduce known difficulties; when the so called inverse crime is avoided, the identification procedure worsens its performance. Filters and smoothing are among different features that deserve further investigation. Although the formulation presented here focuses on the acoustic wave propagation problem, its extension to wave propagation in elastic media is straightforward.

在这项工作中，基于水平集方法的拓扑优化过程被应用于解决声波在时域中传播的反问题。在这类反问题中，必须确定背景介质中是否存在障碍。障碍和背景是通过水平集函数定义的，该函数通过遵循反应扩散方程的解来演化。在这种方法中，与使用汉密尔顿-雅各比方程式时通常会观察到的情况相反，无需对拓扑结构进行初步猜测，也无需进行级别集重新初始化过程。目标函数定义为实验和模拟压力信号之间平方差的域和时间积分。有限元方法用于空间和水平集函数离散化，时间步长过程（Newmark方案）用于解决波传播问题以及灵敏度分析的伴随问题。这两个过程都提供了定义水平集演化的速度场所需的信息。结果表明，所提出的技术能够找到背景介质中障碍物的位置和形状。系统测试表明，正如预期的那样，信号源和接收器的分布对最终解决方案具有重大影响。结果还重现了已知的困难；当所谓的 这两个过程都提供了定义水平集演化的速度场所需的信息。结果表明，所提出的技术能够找到背景介质中障碍物的位置和形状。系统测试表明，正如预期的那样，信号源和接收器的分布对最终解决方案具有重大影响。结果还重现了已知的困难；当所谓的 这两个过程都提供了定义水平集演化的速度场所需的信息。结果表明，所提出的技术能够找到背景介质中障碍物的位置和形状。系统测试表明，正如预期的那样，信号源和接收器的分布对最终解决方案具有重大影响。结果还重现了已知的困难；当所谓的避免了反向犯罪，识别程序会使性能恶化。滤波和平滑是值得进一步研究的不同功能。尽管此处介绍的公式着重于声波传播问题，但它对弹性介质中波传播的扩展很简单。