From bc92cc37c50bf7a6171ab7c8df882d54a4da86d1 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Wed, 20 Aug 2025 18:09:17 -0500
Subject: [PATCH 1/9] update readme

---
 README.md | 93 +++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 91 insertions(+), 2 deletions(-)

diff --git a/README.md b/README.md
index 74e7534..024906b 100644
--- a/README.md
+++ b/README.md
@@ -1,2 +1,91 @@
-# optimization-cloth-tutorial
-A tutorial to implicit cloth simulation with numerical optimization 
+# Implicit Cloth Simulation with Gradient Descent
+
+A gentle introduction to optimization-based implicit time integration for [CSCI 5611](https://umtc.catalog.prod.coursedog.com/courses/8103481).
+
+_This is a work in progress. I released this around 2021 and it hasn't seen many updates. Please let me know if you have suggestions: mattoverby@gmail.com_
+
+## Introduction
+
+Cloth simulation is standard coursework for undergraduate computer graphics.
+Many assignments focus on explicit methods like [forward Euler](https://en.wikipedia.org/wiki/Euler_method).
+While efficient and simple, explicit integrators are notoriously unstable for large time steps and high stiffness.
+Implicit methods, e.g., [backward Euler](https://en.wikipedia.org/wiki/Backward_Euler_method), offer a solution to that problem.
+The standard [Baraff & Witkin approach](https://doi.org/10.1145/280814.280821) (B&W) forms a Taylor approximation to the Eqs. of motion, i.e., Newton's method.
+This results in a block-sparse linear system for which the velocity deltas are solved via [conjugate gradient](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf).
+
+The B&W method is conceptually straight forward. However, there are numerous gotchas not apparent in the original references.
+In particular, working out the Jacobians of some models can be nontrivial. Even when correct, the derivatives can still produce hard-to-debug instabilities.
+The [dynamic deformables course notes](https://doi.org/10.1145/3388769.3407490) describes this well, and I highly recommend reviewing the later chapters on cloth simulation for a thorough review of the topic.
+Nonetheless, implicit time integration remains a valuable and sometimes necessary tool for modern graphics applications.
+
+Recently, optimization-based methods have grown in popularity. This approach formulates the Eqs. of motion as an [iteratively minimized objective function](https://en.wikipedia.org/wiki/Optimization_problem).
+These methods provide unparalleled robustness and extensibility, e.g., [state of the art solvers](https://doi.org/10.1145/3450626.3459767) for cloth simulation.
+The implementation is often easier to construct piece-by-piece than B&W-style integrators.
+
+Below we describe a simple implicit solver for mass-spring systems using gradient descent.
+This is meant solely as a tutorial to optimization-based time stepping, and not for practical applications.
+For that, you'll want to consider higher-order optimization algorithms or [local-global type solvers](https://github.com/alecjacobson/computer-graphics-mass-spring-systems).
+
+## Implementation
+
+The objective we aim to minimize is the sum of internal elastic energy potentials plus a quadratic penalty of linear momentum:
+
+<img src="https://render.githubusercontent.com/render/math?math=\bar{x} = x %2B hv %2B h^2 M^{-1} f_{ext}">, <br/>
+<img src="https://render.githubusercontent.com/render/math?math=g(x) = \frac{1}{2h^2}\|M^{1/2}(x-\bar{x})\|^2 %2B \sum E(x)">,
+
+for vertex locations *x*, time step (sec) *h*, diagonal mass matrix *M*, velocities *v*, and external forces (gravity, wind, etc...)
+<img src="https://render.githubusercontent.com/render/math?math=f_{ext}">. The term <img src="https://render.githubusercontent.com/render/math?math=\bar{x}"> is the *explicit predictor*
+that is computed at the beginning of the time step. Computing a frame of animation amounts to iteratively minimizing the above objective until some convergence criteria is met
+(e.g., the norm of the gradient is below some threshold).
+
+### Interface
+
+Below we lay out the steps to produce an optimization-based solver for implicit cloth simulation.
+The dependencies will be fetched through a [DownloadProject](https://github.com/Crascit/DownloadProject) CMake script.
+[Eigen](https://eigen.tuxfamily.org) is used for linear algebra, which is an industry standard library for vector math.
+Rendering and mesh I/O is handled with [libigl](https://libigl.github.io) which also contains many excellent routines for mesh processing.
+In libigl, meshes are stored as two dense matrices, an nx3 matrix for vertices and an mx3 matrix for faces.
+Animating the mesh is done through a pre-draw callback in which we call our solver to update the vertex positions.
+It takes a little finesse to improve the rendering, so we'll save that for another day.
+
+To compile, execute the following commands:
+1. mkdir build
+2. cd build
+3. cmake ..
+4. make -j
+
+### Energies
+
+Whenever I write a new simulator, I start with the most basic elastic energy as the deformable primitives: a Hookean spring with stiffness *k* and rest length <img src="https://render.githubusercontent.com/render/math?math=\ell">. The potential energy of a spring is <img src="https://render.githubusercontent.com/render/math?math=(k/2)(\|x_a-x_b\|-\ell)^2">.
+These springs are created from the unique edges of tCe triangle mesh, with bending springs created between adjacent triangles.
+The [Cloth class](src/cloth.hpp) generates this data given vertex and face buffers.
+We can also evaluate temporary springs to deal with collisions. One end point is the vertex, and the other is the projection onto the surface it's penetrating.
+
+### Optimization
+
+[Gradient descent](https://en.wikipedia.org/wiki/Gradient_descent) is a simple first-order method that steps along the negative gradient w.r.t. *x* at each iteration: <img src="https://render.githubusercontent.com/render/math?math=p = -\nabla g(x^i)">. We use *i* to denote the iteration and *p* as the descent direction. By all accounts, gradient descent is a poor choice due to its low rate of convergence. You can certainly improve performance with other higher-order optimizers. However, I personally advocate starting with gradient descent when writing new optimization code. The optimizer is so simple it's hard to make a mistake - if there are bugs, they are surely in your energy or gradient calculations. This can be a helpful debugging strategy!
+
+Each iteration, we have to make sure we aren't overshooting the objective and increase the energy. This is accomplished by scaling the descent direction with a scalar *s* using [line search](https://en.wikipedia.org/wiki/Line_search). All together, an iteration of gradient descent involves:
+
+<img src="https://render.githubusercontent.com/render/math?math=p = -\nabla g(x^i)">,<br>
+<img src="https://render.githubusercontent.com/render/math?math=s = linesearch(x^i, p)">,<br>
+<img src="https://render.githubusercontent.com/render/math?math=x^{i %2B 1} = x^i %2B s * p">.
+
+Both energy and gradient use similar calculations so you can save resources (and implementation efforts) by computing them both at same time. I like to use one function for both and skip gradient calculation with a conditional during line search. You can see the evaluation of the objective and gradient in the [Objective class](src/Objective.hpp), with gradient descent in [Solver](src/Solver.hpp).
+
+### Common Problems
+With these pieces you have enough to implement an implicit cloth solver. But there are a number of common problems to be aware of if your solver starts behaving erratically:
+- Using masses that aren't scaled by the area of the mesh
+- The cloth mesh is unrealistically large or small: Each unit of the implementation should correspond to one meter or centimeter. Whatever you choose, be consistent.
+- The time step is too large or small (it should be somewhere between 1/1000 and 1/20)
+- Forgetting to resize or initialize matrix values
+- Dividing by zero and error propogation: A single zero-length spring can produce instabilities, as its result is passed all the way up the food chain to the solver. Using asserts, guards, and std::isfinite any time you do a division will save you a lot of trouble (see the [Objective class](src/Objective.hpp) for an example).
+
+## Next Steps
+
+There are many ways to improve the solver illustrated above:
+- Parallelizing the objective and gradient calculation (I recommend [TBB](https://software.intel.com/en-us/oneapi/onetbb))
+- Collision against triangle meshes with a [signed distance field](https://github.com/InteractiveComputerGraphics/TriangleMeshDistance)
+- Better energy models with triangular elements ([Dynamic Deformables appendix D](https://doi.org/10.1145/3388769.3407490))
+- Preconditioning or accelerated versions of gradient descent
+- Optimizers with better convergence like [L-BFGS](https://en.wikipedia.org/wiki/Limited-memory_BFGS) or [projected Newton](https://doi.org/10.1145/1073368.1073394)

From 178d48de4401a74ee47d9657fa1a2f4820cc23dd Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Wed, 20 Aug 2025 19:16:17 -0500
Subject: [PATCH 2/9] added src

---
 .gitignore          |    1 -
 README.md           |    2 +-
 cmake/libigl.cmake  |   11 +
 data/plane.obj      | 3966 +++++++++++++++++++++++++++++++++++++++++++
 data/sphere.obj     | 2564 ++++++++++++++++++++++++++++
 src/ClothAssert.hpp |   48 +
 src/ClothMesh.hpp   |  140 ++
 src/Objective.hpp   |  108 ++
 src/Solver.hpp      |  133 ++
 src/main.cpp        |   91 +
 10 files changed, 7062 insertions(+), 2 deletions(-)
 create mode 100644 cmake/libigl.cmake
 create mode 100644 data/plane.obj
 create mode 100644 data/sphere.obj
 create mode 100644 src/ClothAssert.hpp
 create mode 100644 src/ClothMesh.hpp
 create mode 100644 src/Objective.hpp
 create mode 100644 src/Solver.hpp
 create mode 100644 src/main.cpp

diff --git a/.gitignore b/.gitignore
index d4fb281..8ea30e8 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,7 +5,6 @@
 *.slo
 *.lo
 *.o
-*.obj
 
 # Precompiled Headers
 *.gch
diff --git a/README.md b/README.md
index 024906b..465159a 100644
--- a/README.md
+++ b/README.md
@@ -58,7 +58,7 @@ To compile, execute the following commands:
 
 Whenever I write a new simulator, I start with the most basic elastic energy as the deformable primitives: a Hookean spring with stiffness *k* and rest length <img src="https://render.githubusercontent.com/render/math?math=\ell">. The potential energy of a spring is <img src="https://render.githubusercontent.com/render/math?math=(k/2)(\|x_a-x_b\|-\ell)^2">.
 These springs are created from the unique edges of tCe triangle mesh, with bending springs created between adjacent triangles.
-The [Cloth class](src/cloth.hpp) generates this data given vertex and face buffers.
+The [ClothMesh class](src/ClothMesh.hpp) generates this data given vertex and face buffers.
 We can also evaluate temporary springs to deal with collisions. One end point is the vertex, and the other is the projection onto the surface it's penetrating.
 
 ### Optimization
diff --git a/cmake/libigl.cmake b/cmake/libigl.cmake
new file mode 100644
index 0000000..1aae978
--- /dev/null
+++ b/cmake/libigl.cmake
@@ -0,0 +1,11 @@
+if(TARGET igl::core)
+    return()
+endif()
+
+include(FetchContent)
+FetchContent_Declare(
+    libigl
+    GIT_REPOSITORY https://github.com/libigl/libigl.git
+    GIT_TAG v2.5.0
+)
+FetchContent_MakeAvailable(libigl)
\ No newline at end of file
diff --git a/data/plane.obj b/data/plane.obj
new file mode 100644
index 0000000..a4f9c26
--- /dev/null
+++ b/data/plane.obj
@@ -0,0 +1,3966 @@
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diff --git a/data/sphere.obj b/data/sphere.obj
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diff --git a/src/ClothAssert.hpp b/src/ClothAssert.hpp
new file mode 100644
index 0000000..7975731
--- /dev/null
+++ b/src/ClothAssert.hpp
@@ -0,0 +1,48 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#ifndef CLOTHASSERT_HPP
+#define CLOTHASSERT_HPP 1
+
+#include <cstdlib> 
+#include <string>
+#include <stdexcept>
+
+namespace cloth
+{
+
+static inline void AssertHandler(bool cond, const std::string& file, const int& line)
+{
+	if (!cond)
+	{
+		std::string err_msg = "Assertion failed in "+file+" line "+std::to_string(line);
+		throw std::runtime_error(err_msg.c_str());
+	}
+}
+
+static inline void AssertHandlerMsg(bool cond, const std::string& file, const int& line, const std::string &msg)
+{
+	if (!cond)
+	{
+		std::string err_msg = "Assertion failed in "+file+" line "+std::to_string(line)+": "+msg;
+		throw std::runtime_error(err_msg.c_str());
+	}
+}
+
+}
+
+// Neat trick that allows macros with multiple arguments:
+// https://stackoverflow.com/questions/3046889/optional-parameters-with-c-macros
+// Also consider using #condition
+
+#define ClothAssert_withmsg(cond, msg) AssertHandlerMsg(cond, std::string(__FILE__), __LINE__, msg)
+#define ClothAssert_nomsg(cond) AssertHandler(cond, std::string(__FILE__), __LINE__)
+#define ClothAssert_stripargs(xx,cond,msg,FUNC, ...) FUNC
+
+#define ClothAssert(...) \
+	ClothAssert_stripargs(,##__VA_ARGS__,\
+	ClothAssert_withmsg(__VA_ARGS__),\
+	ClothAssert_nomsg(__VA_ARGS__),\
+	)
+
+#endif
diff --git a/src/ClothMesh.hpp b/src/ClothMesh.hpp
new file mode 100644
index 0000000..214ab9d
--- /dev/null
+++ b/src/ClothMesh.hpp
@@ -0,0 +1,140 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#ifndef CLOTHMESH_HPP
+#define CLOTHMESH_HPP 1
+
+#include "ClothAssert.hpp"
+#include <Eigen/Dense>
+#include <map>
+
+namespace cloth
+{
+
+class ClothMesh
+{
+public:
+
+	Eigen::MatrixXd V; // rest vertices [nx3]
+	Eigen::MatrixXi F; // faces [fx3]
+	Eigen::MatrixXi E; // edges [ex3]
+	Eigen::VectorXd E0; // edge rest length
+	Eigen::MatrixXi B; // bend edges [bx3]
+	Eigen::VectorXd B0; // bend rest length
+	Eigen::VectorXd masses; // per-vertex masses
+	Eigen::VectorXi V_offset; // per-mesh index into V
+	Eigen::VectorXi F_offset; // per-mesh index into F
+
+	// Adds a mesh to the collection of deformables and returns mesh number
+	inline int add_mesh(const Eigen::MatrixXd &inV, const Eigen::MatrixXi &inF)
+	{
+		int mesh_idx = -1;
+		ClothAssert(inV.rows() > 0 && inV.cols()==3);
+		ClothAssert(inF.rows() > 0 && inF.cols()==3);
+
+		// First mesh?
+		if (V_offset.size()==0)
+		{
+			using namespace Eigen;
+			V_offset = VectorXi::Zero(2);
+			F_offset = VectorXi::Zero(2);
+			V_offset[1] = inV.rows();
+			F_offset[1] = inF.rows();
+			V = inV;
+			F = inF;
+			mesh_idx = 0;
+		}
+		else
+		{
+			mesh_idx = V_offset.size()-1;
+			V_offset.conservativeResize(mesh_idx+2);
+			F_offset.conservativeResize(mesh_idx+2);
+			V_offset[mesh_idx+1] = inV.rows();
+			F_offset[mesh_idx+1] = inF.rows();
+			V.conservativeResize(V.rows() + inV.rows(), 3);
+			V.bottomRows(inV.rows()) = inV;
+			F.conservativeResize(F.rows() + inF.rows(), 3);
+			F.bottomRows(inF.rows()) = inF;
+		}
+
+		masses = Eigen::VectorXd::Constant(V.rows(), 0.1/V.rows());
+
+		compute_edges();
+		return mesh_idx;
+	}
+
+	// Computes edges and cross-edges from faces
+	inline void compute_edges()
+	{
+		using namespace Eigen;
+		std::map<std::pair<int,int>, std::pair<int,int>> edges; // [edge, bend]
+		typedef std::map<std::pair<int,int>, std::pair<int,int>>::iterator EdgeIter;
+		int n_bend = 0; // keep track of valid bend edges (non-border)
+
+		int nf = F.rows();
+		for (int i=0; i<nf; ++i)
+		{
+			for (int j=0; j<3; ++j)
+			{
+				std::pair<int,int> edge = std::make_pair(F(i,j), F(i,(j+1)%3));
+				int adj = F(i,(j+2)%3);
+				if (edge.second < edge.first)
+				{
+					std::swap(edge.first, edge.second);
+				}
+
+				EdgeIter it = edges.find(edge);
+				if (it == edges.end())
+				{
+					edges[edge] = std::make_pair(adj, -1);
+				}
+				else
+				{
+					n_bend++;
+					std::pair<int,int> &adj_pair = it->second;
+					adj_pair.second = adj;
+					if (adj_pair.second < adj_pair.first)
+					{
+						std::swap(adj_pair.first, adj_pair.second);
+					}
+				}
+			}
+		}
+
+		// Loop over map and create eigen buffers
+		E = MatrixXi::Zero(edges.size(), 2);
+		E0 = VectorXd::Constant(edges.size(), -1);
+		B = MatrixXi::Zero(n_bend, 2);
+		B0 = VectorXd::Constant(n_bend, -1);
+
+		int edge_idx = 0;
+		int bend_idx = 0;
+		for(EdgeIter it = edges.begin(); it != edges.end(); ++it, ++edge_idx)
+		{
+			const std::pair<int,int> &e = it->first;
+			E(edge_idx,0) = e.first;
+			E(edge_idx,1) = e.second;
+			E0[edge_idx] = (V.row(e.first) - V.row(e.second)).norm();
+
+			const std::pair<int,int> &b = it->second;
+			if (b.first >= 0 && b.second >= 0)
+			{
+				ClothAssert(bend_idx < B.rows());
+				B(bend_idx,0) = b.first;
+				B(bend_idx,1) = b.second;
+				B0[bend_idx] = (V.row(b.first) - V.row(b.second)).norm();
+				bend_idx++;
+			}
+		}
+
+		// Check to make sure we set every edge
+		ClothAssert(E0.minCoeff() >= 0);
+		ClothAssert(B0.minCoeff() >= 0);
+
+	} // end compute edges
+
+};
+
+}
+
+#endif
diff --git a/src/Objective.hpp b/src/Objective.hpp
new file mode 100644
index 0000000..4e0d1ff
--- /dev/null
+++ b/src/Objective.hpp
@@ -0,0 +1,108 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#ifndef OBJECTIVE_HPP
+#define OBJECTIVE_HPP 1
+
+#include "ClothAssert.hpp"
+#include "ClothMesh.hpp"
+
+namespace cloth
+{
+
+class Objective
+{
+public:
+
+	// Computes and returns the Hookean energy of a spring with rest length r and stiffness k.
+	// If x.rows == grad.rows, the gradients (f, -f) are added to grad.
+	double spring_gradient(
+		const Eigen::MatrixXd &x,
+		int e0,
+		int e1,
+		double r,
+		double k,
+		Eigen::MatrixXd &grad) const
+	{
+		using namespace Eigen;
+		Vector3d x0 = x.row(e0);
+		Vector3d x1 = x.row(e1);
+		Vector3d edge = (x0-x1);
+		double l = edge.norm();
+		if (grad.rows() == x.rows() && l > 1e-12)
+		{
+			edge /= l;
+			edge *= (l-r)*k;
+			ClothAssert(edge.allFinite());
+			grad.row(e0) += edge;
+			grad.row(e1) -= edge;
+		}
+		double energy = 0.5 * k * (l-r) * (l-r); // (k/2)||l-l0||^2
+		ClothAssert(std::isfinite(energy));
+		return  energy;
+	}
+
+	// Spring energies that push vertices out of collision.
+	// We'll cheat a bit and hard code the sphere radius and center since we know it.
+	double collision_gradient(
+		const Eigen::MatrixXd &x,
+		int idx,
+		double k,
+		Eigen::MatrixXd &grad) const
+	{
+		using namespace Eigen;
+		Vector3d cent(0,0,0);
+		double rad = 0.51;
+		Vector3d xi = x.row(idx);
+		Vector3d dir = (xi-cent);
+		double l = dir.norm();
+		if (l > rad) { return 0; }
+		if (grad.rows() == x.rows() && l > 1e-12)
+		{
+			dir /= l;
+			dir *= (l-rad)*k;
+			ClothAssert(dir.allFinite());
+			grad.row(idx) += dir;
+		}
+		double energy = 0.5 * k * (l-rad) * (l-rad); // (k/2)||l-l0||^2
+		ClothAssert(std::isfinite(energy));
+		return energy;
+	}
+
+	// Momentum potential: ||x-xbar||^2_M / (2dt^2)
+	// Like springs above, if xrows == grad.rows, gradient is added.
+	// Unlike springs, the momentum is a global term.
+	inline double momentum_gradient(
+		const Eigen::MatrixXd &x,
+		const Eigen::MatrixXd &x_start,
+		const Eigen::MatrixXd &v,
+		const Eigen::VectorXd &masses,
+		double dt,
+		Eigen::MatrixXd &grad) const
+	{
+		using namespace Eigen;
+		// Some of this can be precomputed to speed up evaluations
+		MatrixXd x_bar = x_start + dt * v;
+		MatrixXd dx = x - x_bar;
+		MatrixXd M_dx = masses.asDiagonal() * dx;
+		double dot = 0;
+		for (int i=0; i<3; ++i)
+		{
+			dot += dx.col(i).dot(M_dx.col(i));
+		}
+
+		double energy = dot / (2.0*dt*dt);
+		if (grad.rows() == x.rows())
+		{
+			M_dx *= (1.0 / (dt*dt));
+			grad += M_dx;
+		}
+		ClothAssert(std::isfinite(energy));
+		return energy;
+	}
+
+};
+
+}
+
+#endif
diff --git a/src/Solver.hpp b/src/Solver.hpp
new file mode 100644
index 0000000..1b6af68
--- /dev/null
+++ b/src/Solver.hpp
@@ -0,0 +1,133 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#ifndef SOLVER_HPP
+#define SOLVER_HPP 1
+
+#include "Objective.hpp"
+#include "ClothMesh.hpp"
+
+namespace cloth
+{
+
+class Solver
+{
+public:
+
+	// State variables
+	Eigen::MatrixXd x; // positions [nx3]
+	Eigen::MatrixXd v; // velocities [nx3]
+	Eigen::MatrixXd x_start; // start of timestep [nx3]
+	double spring_k;
+	double bend_k;
+	double collision_k;
+
+	Solver() :
+		spring_k(100),
+		bend_k(100),
+		collision_k(300)
+		{}
+
+	// Initalizes state variables, returns true if success
+	bool initialize(const ClothMesh &cloth)
+	{
+		if (cloth.V.rows() == 0 || cloth.V.cols() != 3)
+		{
+			return false;
+		}
+
+		x = cloth.V;
+		v = Eigen::MatrixXd::Zero(x.rows(), 3);
+		x_start = x;
+		return true;
+	}
+
+	// Computes the energy and gradient (if desired) of the objective
+	double gradient(const ClothMesh &cloth, const Objective &objective,
+		double dt, const Eigen::MatrixXd &xk, Eigen::MatrixXd &grad) const
+	{
+		double tot_energy = 0;
+
+		// Momentum potential
+		tot_energy += objective.momentum_gradient(
+			xk, x_start, v, cloth.masses, dt, grad);
+
+		// Stretch springs
+		int num_edges = cloth.E.rows();
+		for (int i=0; i<num_edges; ++i)
+		{
+			Eigen::RowVector2i e = cloth.E.row(i);
+			tot_energy += objective.spring_gradient(
+				xk, e[0], e[1], cloth.E0[i], spring_k, grad);
+		}
+
+		// Linear bend springs
+		int num_bend = cloth.B.rows();
+		for (int i=0; i<num_bend; ++i)
+		{
+			Eigen::RowVector2i b = cloth.B.row(i);
+			tot_energy += objective.spring_gradient(
+				xk, b[0], b[1], cloth.B0[i], bend_k, grad);
+		}
+
+		// Collision springs
+		int num_verts = xk.rows();
+		for (int i=0; i<num_verts; ++i)
+		{
+			tot_energy += objective.collision_gradient(
+				xk, i, collision_k, grad);
+		}
+
+		return tot_energy;
+	}
+
+	double energy(const ClothMesh &cloth, const Objective &objective, double dt, const Eigen::MatrixXd &xk) const
+	{
+		Eigen::MatrixXd dummy; // grad computation ignored if grad.size != x.size
+		return gradient(cloth, objective, dt, xk, dummy);
+	}
+
+	void solve(const ClothMesh &cloth, const Objective &objective, double dt)
+	{
+		using namespace Eigen;
+		ClothAssert(x.rows() == cloth.V.rows() && x.cols() == 3);
+		ClothAssert(v.rows() == x.rows() && v.cols() == x.cols());
+
+		x_start = x; // cache start state
+		v.col(1).array() += dt * -9.8; // gravity
+		x = x_start + dt * v; // initial guess
+		MatrixXd grad = MatrixXd::Zero(x.rows(), 3);
+		MatrixXd xk = MatrixXd::Zero(x.rows(), 3);
+
+		int max_iters = 400;
+		for (int iter = 0; iter < max_iters; ++iter)
+		{
+			// Compute current energy and gradient
+			xk = x; // store state at current iteration
+			grad.setZero();
+			double energy_k = gradient(cloth, objective, dt, x, grad);
+
+			// Check for convergence
+			if (grad.norm() < 0.01)
+			{
+				break;
+			}
+
+			// Take a full step, i.e. alpha=1
+			x = xk - grad;
+
+			// Line search until objective decreased
+			double alpha = 1;
+			while (energy(cloth, objective, dt, x) > energy_k)
+			{
+				alpha *= 0.5;
+				x = xk - alpha * grad;
+			}
+		}
+	}
+
+};
+
+}
+
+#endif
diff --git a/src/main.cpp b/src/main.cpp
new file mode 100644
index 0000000..2c54eff
--- /dev/null
+++ b/src/main.cpp
@@ -0,0 +1,91 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#include <igl/opengl/glfw/Viewer.h>
+#include <igl/readOBJ.h>
+#include <iostream>
+#include <Eigen/Geometry>
+
+#include "Solver.hpp"
+
+using namespace Eigen;
+using namespace cloth;
+
+int main(int argc, char *argv[])
+{
+	(void)(argc);
+	(void)(argv);
+
+	std::string plane = CLOTH_ROOT_DIR "/data/plane.obj";
+	std::string sphere = CLOTH_ROOT_DIR "/data/sphere.obj";
+	MatrixXd V, cV;
+	MatrixXi F, cF;
+	if (!igl::readOBJ(plane, V, F) || !igl::readOBJ(sphere, cV, cF))
+	{
+		return EXIT_FAILURE;
+	}
+
+	// Shrink the sphere (radius 1, centered about origin)
+	//cV *= 0.75;
+
+	// Move cloth above the sphere
+	V.col(1).array() += 0.5;
+
+	// Create cloth object
+	ClothMesh cloth;
+	cloth.add_mesh(V, F);
+
+	// Objective function
+	Objective objective;
+
+	// Create solver
+	double dt = 1.0 / 24.0;
+	Solver solver;
+	if (!solver.initialize(cloth))
+	{
+		return EXIT_FAILURE;
+	}
+
+	// Create combined matrices to pass to igl
+	MatrixXd fullX(V.rows() + cV.rows(), 3);
+	fullX.topRows(V.rows()) = solver.x;
+	fullX.bottomRows(cV.rows()) = cV;
+	MatrixXi fullF(F.rows() + cF.rows(), 3);
+	fullF.topRows(F.rows()) = F;
+	fullF.bottomRows(cF.rows()) = cF;
+	fullF.bottomRows(cF.rows()).array() += V.rows();
+
+	std::cout << "Press A to simulate, R to reset" << std::endl;
+
+	// Create viewer
+	igl::opengl::glfw::Viewer viewer;
+	viewer.data().set_face_based(true);
+	viewer.data().set_mesh(fullX, fullF);
+	viewer.core().is_animating = false;
+
+	viewer.callback_key_pressed = [&](igl::opengl::glfw::Viewer&, unsigned int key, int)->bool
+	{
+		char c = char(key);
+		if (c == 'r')
+		{
+			solver.initialize(cloth);
+		}
+		return false;
+	};
+
+	viewer.callback_pre_draw = [&](igl::opengl::glfw::Viewer&)->bool
+	{
+		if (viewer.core().is_animating)
+		{
+			solver.solve(cloth, objective, dt); // solve the time step
+			fullX.topRows(V.rows()) = solver.x; // update vertex positions
+			viewer.data().set_mesh(fullX, fullF);
+			viewer.data().compute_normals(); // update normals after defo
+		}
+		return false;
+	};
+
+	viewer.launch();
+
+	return EXIT_SUCCESS;
+}

From 5cfbff076db028ec40e9ba34d9e6a206c86c7192 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Wed, 20 Aug 2025 20:21:28 -0500
Subject: [PATCH 3/9] added test

---
 src/main.cpp  | 30 ++++++++++++----------------
 test/test.cpp | 54 +++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 66 insertions(+), 18 deletions(-)
 create mode 100644 test/test.cpp

diff --git a/src/main.cpp b/src/main.cpp
index 2c54eff..a064e0f 100644
--- a/src/main.cpp
+++ b/src/main.cpp
@@ -5,7 +5,6 @@
 #include <igl/readOBJ.h>
 #include <iostream>
 #include <Eigen/Geometry>
-
 #include "Solver.hpp"
 
 using namespace Eigen;
@@ -18,9 +17,9 @@ int main(int argc, char *argv[])
 
 	std::string plane = CLOTH_ROOT_DIR "/data/plane.obj";
 	std::string sphere = CLOTH_ROOT_DIR "/data/sphere.obj";
-	MatrixXd V, cV;
-	MatrixXi F, cF;
-	if (!igl::readOBJ(plane, V, F) || !igl::readOBJ(sphere, cV, cF))
+	MatrixXd V, sphereV;
+	MatrixXi F, sphereF;
+	if (!igl::readOBJ(plane, V, F) || !igl::readOBJ(sphere, sphereV, sphereF))
 	{
 		return EXIT_FAILURE;
 	}
@@ -29,7 +28,7 @@ int main(int argc, char *argv[])
 	//cV *= 0.75;
 
 	// Move cloth above the sphere
-	V.col(1).array() += 0.5;
+	V.col(1).array() += 0.6;
 
 	// Create cloth object
 	ClothMesh cloth;
@@ -46,21 +45,16 @@ int main(int argc, char *argv[])
 		return EXIT_FAILURE;
 	}
 
-	// Create combined matrices to pass to igl
-	MatrixXd fullX(V.rows() + cV.rows(), 3);
-	fullX.topRows(V.rows()) = solver.x;
-	fullX.bottomRows(cV.rows()) = cV;
-	MatrixXi fullF(F.rows() + cF.rows(), 3);
-	fullF.topRows(F.rows()) = F;
-	fullF.bottomRows(cF.rows()) = cF;
-	fullF.bottomRows(cF.rows()).array() += V.rows();
-
 	std::cout << "Press A to simulate, R to reset" << std::endl;
 
 	// Create viewer
 	igl::opengl::glfw::Viewer viewer;
 	viewer.data().set_face_based(true);
-	viewer.data().set_mesh(fullX, fullF);
+	viewer.data().set_mesh(V, F);
+	viewer.data().set_colors(RowVector3d(0.659, 0.847, 1));
+	viewer.append_mesh();
+	viewer.data(1).set_mesh(sphereV, sphereF);
+	viewer.data(1).set_colors(RowVector3d(0.7, 0.7, 0.7));
 	viewer.core().is_animating = false;
 
 	viewer.callback_key_pressed = [&](igl::opengl::glfw::Viewer&, unsigned int key, int)->bool
@@ -78,9 +72,9 @@ int main(int argc, char *argv[])
 		if (viewer.core().is_animating)
 		{
 			solver.solve(cloth, objective, dt); // solve the time step
-			fullX.topRows(V.rows()) = solver.x; // update vertex positions
-			viewer.data().set_mesh(fullX, fullF);
-			viewer.data().compute_normals(); // update normals after defo
+			V = solver.x; // update vertex positions
+			viewer.data(0).set_mesh(V, F);
+			viewer.data(0).compute_normals(); // update normals after defo
 		}
 		return false;
 	};
diff --git a/test/test.cpp b/test/test.cpp
new file mode 100644
index 0000000..85b26cc
--- /dev/null
+++ b/test/test.cpp
@@ -0,0 +1,54 @@
+// Copyright Matt Overby 2021.
+// Distributed under the MIT License.
+
+#include <igl/readOBJ.h>
+#include <iostream>
+#include <Eigen/Geometry>
+#include "Solver.hpp"
+
+using namespace Eigen;
+using namespace cloth;
+
+int main(int argc, char *argv[])
+{
+	(void)(argc);
+	(void)(argv);
+
+	std::string plane = CLOTH_ROOT_DIR "/data/plane.obj";
+	std::string sphere = CLOTH_ROOT_DIR "/data/sphere.obj";
+	MatrixXd V, cV;
+	MatrixXi F, cF;
+	if (!igl::readOBJ(plane, V, F) || !igl::readOBJ(sphere, cV, cF))
+	{
+		return EXIT_FAILURE;
+	}
+
+	// Shrink the sphere (radius 1, centered about origin)
+	//cV *= 0.75;
+
+	// Move cloth above the sphere
+	V.col(1).array() += 0.5;
+
+	// Create cloth object
+	ClothMesh cloth;
+	cloth.add_mesh(V, F);
+
+	// Objective function
+	Objective objective;
+
+	// Create solver
+	double dt = 1.0 / 24.0;
+	Solver solver;
+	if (!solver.initialize(cloth))
+	{
+		return EXIT_FAILURE;
+	}
+
+    int timesteps = 10;
+    for (int step=0; step < timesteps; ++step)
+    {
+        solver.solve(cloth, objective, dt); // solve the time step
+    }
+
+	return EXIT_SUCCESS;
+}

From 087d736be2b7e4e849dbd2be6b402b5b4d0cb803 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Wed, 20 Aug 2025 20:34:19 -0500
Subject: [PATCH 4/9] update test

---
 src/Solver.hpp | 7 ++++---
 src/main.cpp   | 3 ---
 test/test.cpp  | 8 +++++---
 3 files changed, 9 insertions(+), 9 deletions(-)

diff --git a/src/Solver.hpp b/src/Solver.hpp
index 1b6af68..1b065a3 100644
--- a/src/Solver.hpp
+++ b/src/Solver.hpp
@@ -21,11 +21,13 @@ class Solver
 	double spring_k;
 	double bend_k;
 	double collision_k;
+	int max_solver_iter;
 
 	Solver() :
 		spring_k(100),
 		bend_k(100),
-		collision_k(300)
+		collision_k(300),
+		max_solver_iter(400)
 		{}
 
 	// Initalizes state variables, returns true if success
@@ -99,8 +101,7 @@ class Solver
 		MatrixXd grad = MatrixXd::Zero(x.rows(), 3);
 		MatrixXd xk = MatrixXd::Zero(x.rows(), 3);
 
-		int max_iters = 400;
-		for (int iter = 0; iter < max_iters; ++iter)
+		for (int iter = 0; iter < max_solver_iter; ++iter)
 		{
 			// Compute current energy and gradient
 			xk = x; // store state at current iteration
diff --git a/src/main.cpp b/src/main.cpp
index a064e0f..2b04ccd 100644
--- a/src/main.cpp
+++ b/src/main.cpp
@@ -24,9 +24,6 @@ int main(int argc, char *argv[])
 		return EXIT_FAILURE;
 	}
 
-	// Shrink the sphere (radius 1, centered about origin)
-	//cV *= 0.75;
-
 	// Move cloth above the sphere
 	V.col(1).array() += 0.6;
 
diff --git a/test/test.cpp b/test/test.cpp
index 85b26cc..5a6bd9a 100644
--- a/test/test.cpp
+++ b/test/test.cpp
@@ -23,9 +23,6 @@ int main(int argc, char *argv[])
 		return EXIT_FAILURE;
 	}
 
-	// Shrink the sphere (radius 1, centered about origin)
-	//cV *= 0.75;
-
 	// Move cloth above the sphere
 	V.col(1).array() += 0.5;
 
@@ -39,6 +36,11 @@ int main(int argc, char *argv[])
 	// Create solver
 	double dt = 1.0 / 24.0;
 	Solver solver;
+
+	// 20 iters won't converge but it will test
+	// all the functions are operational.
+	solver.max_solver_iter = 20;
+	
 	if (!solver.initialize(cloth))
 	{
 		return EXIT_FAILURE;

From ac599ee2c32424d7ca731660b35df3205bb28c10 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Thu, 21 Aug 2025 11:27:32 -0500
Subject: [PATCH 5/9] updated github action

---
 .github/workflows/build_and_test.yml | 1 +
 1 file changed, 1 insertion(+)

diff --git a/.github/workflows/build_and_test.yml b/.github/workflows/build_and_test.yml
index c244123..f8969bd 100644
--- a/.github/workflows/build_and_test.yml
+++ b/.github/workflows/build_and_test.yml
@@ -21,6 +21,7 @@ jobs:
 
     - name: build
       run: |
+        cd optimization-cloth-tutorial
         cmake -DCLOTH_DEBUG=ON .
         make -j
 

From faffef56a49c86b1bd36434da2ee293410872ec7 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Thu, 21 Aug 2025 11:28:36 -0500
Subject: [PATCH 6/9] updated github action

---
 .github/workflows/build_and_test.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/build_and_test.yml b/.github/workflows/build_and_test.yml
index f8969bd..c723464 100644
--- a/.github/workflows/build_and_test.yml
+++ b/.github/workflows/build_and_test.yml
@@ -20,8 +20,8 @@ jobs:
         sudo apt-get install -y build-essential cmake
 
     - name: build
+      working-directory: ${{ github.workspace }}/optimization-cloth-tutorial
       run: |
-        cd optimization-cloth-tutorial
         cmake -DCLOTH_DEBUG=ON .
         make -j
 

From 467a7ab181211efbff8e97cdf2ff6defa67dc852 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Thu, 21 Aug 2025 11:34:13 -0500
Subject: [PATCH 7/9] updated github action

---
 .github/workflows/build_and_test.yml | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/.github/workflows/build_and_test.yml b/.github/workflows/build_and_test.yml
index c723464..b3281c5 100644
--- a/.github/workflows/build_and_test.yml
+++ b/.github/workflows/build_and_test.yml
@@ -20,10 +20,11 @@ jobs:
         sudo apt-get install -y build-essential cmake
 
     - name: build
-      working-directory: ${{ github.workspace }}/optimization-cloth-tutorial
+      working-directory: ${{ github.workspace }} 
       run: |
         cmake -DCLOTH_DEBUG=ON .
         make -j
 
     - name: test
+      working-directory: ${{ github.workspace }} 
       run: make test

From 43f1e8503dc22ae6cef15a0327bbf9ff9787def3 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Thu, 21 Aug 2025 12:49:02 -0500
Subject: [PATCH 8/9] added cmakelists

---
 CMakeLists.txt | 42 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 42 insertions(+)
 create mode 100644 CMakeLists.txt

diff --git a/CMakeLists.txt b/CMakeLists.txt
new file mode 100644
index 0000000..f969716
--- /dev/null
+++ b/CMakeLists.txt
@@ -0,0 +1,42 @@
+# Copyright Matt Overby 2021.
+# Distributed under the MIT License.
+cmake_minimum_required(VERSION 3.5)
+
+project(cloth_sim_tutorial LANGUAGES CXX)
+
+option(CLOTH_DEBUG "run with debug options" OFF)
+set(CMAKE_CXX_STANDARD 14)
+set(CMAKE_CXX_STANDARD_REQUIRED ON)
+list(PREPEND CMAKE_MODULE_PATH ${CMAKE_CURRENT_SOURCE_DIR}/cmake)
+add_definitions(-DCLOTH_ROOT_DIR="${CMAKE_CURRENT_SOURCE_DIR}")
+
+# Change compile flags for debug
+if (CLOTH_DEBUG)
+	add_definitions(-DCLOTH_DEBUG)
+	set(CMAKE_BUILD_TYPE Debug)
+	if(CMAKE_COMPILER_IS_GNUCC OR CMAKE_COMPILER_IS_GNUCXX)
+		set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -Wall -Wundef -Wextra -Wno-long-long -Wlogical-op -Wshadow -Wmaybe-uninitialized")
+	endif()
+else()
+	set(CMAKE_BUILD_TYPE Release)
+endif()
+
+# Libigl also includes Eigen, etc
+include(libigl)
+igl_include(glfw)
+
+# Gather src
+include_directories(${CMAKE_CURRENT_SOURCE_DIR}/src)
+file(GLOB SRC
+	${CMAKE_CURRENT_SOURCE_DIR}/src/*.hpp
+	${CMAKE_CURRENT_SOURCE_DIR}/src/*.cpp)
+
+# Create binary
+add_executable(${PROJECT_NAME} ${SRC})
+target_link_libraries(${PROJECT_NAME} PUBLIC igl::glfw)
+
+# Unit testing (work in progress)
+enable_testing()
+add_executable(cloth_test ${CMAKE_CURRENT_SOURCE_DIR}/test/test.cpp)
+target_link_libraries(cloth_test PUBLIC igl::glfw)
+add_test(NAME TestClothSolver COMMAND cloth_test)

From 35d10af6f77fe2c92a1c5aad2c6d19ca359c12c0 Mon Sep 17 00:00:00 2001
From: mattoverby <mattoverby@gmail.com>
Date: Thu, 21 Aug 2025 12:55:33 -0500
Subject: [PATCH 9/9] cloth test only flag

---
 .github/workflows/build_and_test.yml |  2 +-
 CMakeLists.txt                       | 12 ++++++++----
 2 files changed, 9 insertions(+), 5 deletions(-)

diff --git a/.github/workflows/build_and_test.yml b/.github/workflows/build_and_test.yml
index b3281c5..9514767 100644
--- a/.github/workflows/build_and_test.yml
+++ b/.github/workflows/build_and_test.yml
@@ -22,7 +22,7 @@ jobs:
     - name: build
       working-directory: ${{ github.workspace }} 
       run: |
-        cmake -DCLOTH_DEBUG=ON .
+        cmake -DCLOTH_TESTING_ONLY=ON .
         make -j
 
     - name: test
diff --git a/CMakeLists.txt b/CMakeLists.txt
index f969716..40242ff 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -5,6 +5,8 @@ cmake_minimum_required(VERSION 3.5)
 project(cloth_sim_tutorial LANGUAGES CXX)
 
 option(CLOTH_DEBUG "run with debug options" OFF)
+option(CLOTH_TESTING_ONLY "only compile unit tests" OFF)
+
 set(CMAKE_CXX_STANDARD 14)
 set(CMAKE_CXX_STANDARD_REQUIRED ON)
 list(PREPEND CMAKE_MODULE_PATH ${CMAKE_CURRENT_SOURCE_DIR}/cmake)
@@ -23,7 +25,6 @@ endif()
 
 # Libigl also includes Eigen, etc
 include(libigl)
-igl_include(glfw)
 
 # Gather src
 include_directories(${CMAKE_CURRENT_SOURCE_DIR}/src)
@@ -32,11 +33,14 @@ file(GLOB SRC
 	${CMAKE_CURRENT_SOURCE_DIR}/src/*.cpp)
 
 # Create binary
-add_executable(${PROJECT_NAME} ${SRC})
-target_link_libraries(${PROJECT_NAME} PUBLIC igl::glfw)
+if (NOT CLOTH_TESTING_ONLY)
+	igl_include(glfw)
+	add_executable(${PROJECT_NAME} ${SRC})
+	target_link_libraries(${PROJECT_NAME} PUBLIC igl::glfw)
+endif()
 
 # Unit testing (work in progress)
 enable_testing()
 add_executable(cloth_test ${CMAKE_CURRENT_SOURCE_DIR}/test/test.cpp)
-target_link_libraries(cloth_test PUBLIC igl::glfw)
+target_link_libraries(cloth_test PUBLIC igl::core)
 add_test(NAME TestClothSolver COMMAND cloth_test)