In this paper, we study the dynamics of rigid particles in either viscoelastic flow or unsteady Stokes flow in 3D. We start with the Cauchy momentum equation in time domain, and apply the Fourier transform to recast the problem in frequency domain as a Brinkman equation with imaginary coefficients. The Fourier space equations are reformulated in terms of boundary integrals. This formulation allows us to compute the velocity of the particles if the force is given (forward problem), and to compute the force when the velocity is given (inverse problem). We then propose a special mapping scheme such that the singularity of the surface integrals can be removed and a higher order quadrature can be established. Using a two-particle system, we demonstrate the order of convergence of our algorithms and validate our numerical results with known analytic or asymptotic solutions. Numerical experiments reveal that our method is formally high-order accurate and able to tackle problems with closely packed particles.

在本文中，我们研究了3D中粘弹性流或不稳定Stokes流中刚性颗粒的动力学。我们从时域的柯西动量方程开始，然后应用傅立叶变换将频域中的问题重铸为具有虚系数的布林克曼方程。傅里叶空间方程根据边界积分重新公式化。这种公式使我们能够在给定力的情况下计算粒子的速度（正向问题），而在给定速度的情况下计算粒子的速度（反问题）。然后，我们提出一种特殊的映射方案，以便可以去除表面积分的奇异性，并可以建立更高阶的正交。使用两粒子系统 我们演示了算法收敛的顺序，并用已知的解析或渐近解验证了我们的数值结果。数值实验表明，我们的方法在形式上是高阶准确的，并且能够解决紧密堆积的粒子的问题。